feat(Algorithms): comparison sort lower bound via decision-tree induction

Cslib.lean

@@ -2,6 +2,7 @@ module  -- shake: keep-all --deprecated_module: ignore
 
 public import Cslib.Algorithms.Lean.MergeSort.MergeSort
 public import Cslib.Algorithms.Lean.TimeM
+public import Cslib.Algorithms.LowerBounds.ComparisonSort
 public import Cslib.Computability.Automata.Acceptors.Acceptor
 public import Cslib.Computability.Automata.Acceptors.OmegaAcceptor
 public import Cslib.Computability.Automata.DA.Basic

Cslib/Algorithms/LowerBounds/ComparisonSort.lean

@@ -0,0 +1,232 @@
+/-
+Copyright (c) 2026 Samuel Schlesinger. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Samuel Schlesinger
+-/
+
+module
+
+public import Cslib.Foundations.Control.Monad.Free
+public import Cslib.Algorithms.Lean.TimeM
+public import Mathlib.Data.Finset.Basic
+public import Mathlib.Data.Finset.Lattice.Fold
+public import Mathlib.Data.Fintype.Perm
+public import Mathlib.Data.Nat.Factorial.Basic
+public import Mathlib.Data.Nat.Log
+
+/-!
+# The comparison sort lower bound
+
+This file proves the `Ω(n log n)` worst-case lower bound on comparison sorting.
+
+A program `P : FreeM (SortOps α) β` is a binary decision tree: `pure b` is a leaf, and
+`(FreeM.lift (SortOps.cmpLE x y)).bind cont` is a node with children `cont true` and
+`cont false`. `run P le` and `cost P le` give the result and the number of comparisons
+of `P` under a comparator `le`; they are the two projections of the canonical
+interpretation (`FreeM.liftM`) of `P` into `TimeM ℕ`, and so satisfy `run_bind` and
+`cost_bind`. `worstTime P` is the worst case of `cost` over the orders `permLE σ`
+induced by permutations `σ` of `Fin n`.
+
+By structural induction on `P`, `card_image_run_le_two_pow` bounds the number of
+distinct results over any finite family of comparators by `2 ^ c`, where `c` is the
+worst-case number of comparisons over that family. A program that sorts under every
+`permLE σ` has `n !` distinct outputs, so `n ! ≤ 2 ^ worstTime P`.
+
+## Main results
+
+- `card_image_run_le_two_pow`: a program attains at most `2 ^ c` distinct results over a
+  finite comparator family with worst-case cost `c`.
+- `factorial_le_two_pow_worstTime`: a program sorting all hidden orders on `Fin n`
+  satisfies `n ! ≤ 2 ^ worstTime P`.
+- `log_factorial_le_worstTime`: `Nat.log 2 (n !) ≤ worstTime P`.
+- `cmpSort_lower_bound`: `n / 2 * Nat.log 2 (n / 2) ≤ worstTime P`.
+
+## Tags
+
+sorting, lower bound, decision tree, query complexity, free monad
+-/
+
+@[expose] public section
+
+namespace Cslib.Algorithms
+
+open scoped Nat
+
+/--
+A query type for comparison-based sorting, with a single query that compares two elements.
+-/
+inductive SortOps.{u} (α : Type u) : Type → Type _ where
+  /-- `cmpLE x y` is intended to return `true` if `x ≤ y` and `false` otherwise.
+  The specific order relation depends on the comparator the program is run against. -/
+  | cmpLE (x : α) (y : α) : SortOps α Bool
+
+variable {α : Type u} {β γ : Type}
+
+open Cslib.Algorithms.Lean in
+/-- Interpret the comparison query into `TimeM ℕ`: answer with `le x y`, at cost `1`. -/
+def sortHandler (le : α → α → Bool) : {ι : Type} → SortOps α ι → TimeM ℕ ι
+  | _, .cmpLE x y => ⟨le x y, 1⟩
+
+/-- The result of running a comparison program against a comparator. -/
+def run (P : FreeM (SortOps α) β) (le : α → α → Bool) : β :=
+  (P.liftM (sortHandler le)).ret
+
+/-- The number of comparisons a program performs against a comparator. -/
+def cost (P : FreeM (SortOps α) β) (le : α → α → Bool) : ℕ :=
+  (P.liftM (sortHandler le)).time
+
+@[simp, grind =]
+lemma run_pure (b : β) (le : α → α → Bool) : run (pure b) le = b := rfl
+
+@[simp, grind =]
+lemma run_lift (x y : α) (le : α → α → Bool) :
+    run (FreeM.lift (SortOps.cmpLE x y)) le = le x y := rfl
+
+@[simp, grind =]
+lemma run_lift_bind (x y : α) (cont : Bool → FreeM (SortOps α) β) (le : α → α → Bool) :
+    run ((FreeM.lift (SortOps.cmpLE x y)).bind cont) le = run (cont (le x y)) le := rfl
+
+@[simp, grind =]
+lemma run_bind (P : FreeM (SortOps α) β) (f : β → FreeM (SortOps α) γ)
+    (le : α → α → Bool) :
+    run (P >>= f) le = run (f (run P le)) le := by
+  simp [run, FreeM.liftM_bind]
+
+@[simp, grind =]
+lemma cost_pure (b : β) (le : α → α → Bool) :
+    cost (pure b : FreeM (SortOps α) β) le = 0 := rfl
+
+@[simp, grind =]
+lemma cost_lift (x y : α) (le : α → α → Bool) :
+    cost (FreeM.lift (SortOps.cmpLE x y)) le = 1 := rfl
+
+@[simp, grind =]
+lemma cost_lift_bind (x y : α) (cont : Bool → FreeM (SortOps α) β) (le : α → α → Bool) :
+    cost ((FreeM.lift (SortOps.cmpLE x y)).bind cont) le = 1 + cost (cont (le x y)) le := rfl
+
+@[simp, grind =]
+lemma cost_bind (P : FreeM (SortOps α) β) (f : β → FreeM (SortOps α) γ)
+    (le : α → α → Bool) :
+    cost (P >>= f) le = cost P le + cost (f (run P le)) le := by
+  simp [run, cost, FreeM.liftM_bind]
+
+/--
+A comparison program making at most `t` comparisons against every comparator in a finite
+family `S` attains at most `2 ^ t` distinct results over `S`: a binary decision tree of
+depth `t` has at most `2 ^ t` leaves.
+-/
+theorem card_image_run_le_two_pow_of_cost_le [DecidableEq β]
+    {P : FreeM (SortOps α) β} {S : Finset (α → α → Bool)} {t : ℕ}
+    (ht : ∀ le ∈ S, cost P le ≤ t) :
+    (S.image fun le => run P le).card ≤ 2 ^ t := by
+  classical
+  induction P generalizing S t with
+  | pure b =>
+    exact (Finset.card_le_card (Finset.image_subset_iff.2 fun le _ =>
+      Finset.mem_singleton_self b)).trans (by simpa using Nat.one_le_two_pow)
+  | lift_bind op cont ih =>
+    cases op with
+    | cmpLE x y =>
+      rcases S.eq_empty_or_nonempty with rfl | ⟨le₁, hle₁⟩
+      · simp
+      obtain ⟨t, rfl⟩ : ∃ t', t = t' + 1 := ⟨t - 1, by grind [ht le₁ hle₁]⟩
+      set St := S.filter (fun le => le x y = true) with hSt
+      set Sf := S.filter (fun le => ¬le x y = true) with hSf
+      -- Split the image along the answer to the root comparison.
+      have himage : (S.image fun le => run ((FreeM.lift (SortOps.cmpLE x y)).bind cont) le) =
+          (St.image fun le => run (cont true) le) ∪
+          (Sf.image fun le => run (cont false) le) := by
+        rw [← Finset.filter_union_filter_not_eq (p := fun le => le x y = true) S,
+          Finset.image_union, hSt, hSf]
+        congr 1
+        · exact Finset.image_congr fun le hle => by grind
+        · exact Finset.image_congr fun le hle => by grind
+      -- Each branch has cost at most `t` over its part of the family.
+      have h₁ : (St.image fun le => run (cont true) le).card ≤ 2 ^ t :=
+        ih true fun le hle => by grind [ht le]
+      have h₂ : (Sf.image fun le => run (cont false) le).card ≤ 2 ^ t :=
+        ih false fun le hle => by grind [ht le]
+      calc (S.image fun le => run ((FreeM.lift (SortOps.cmpLE x y)).bind cont) le).card
+          ≤ (St.image fun le => run (cont true) le).card +
+            (Sf.image fun le => run (cont false) le).card :=
+            himage ▸ Finset.card_union_le _ _
+        _ ≤ 2 ^ t + 2 ^ t := Nat.add_le_add h₁ h₂
+        _ = 2 ^ (t + 1) := by grind
+
+/--
+Over a finite family `S` of comparators, a comparison program attains at most `2 ^ c`
+distinct results, where `c` is its worst-case number of comparisons over `S`.
+-/
+theorem card_image_run_le_two_pow [DecidableEq β]
+    (P : FreeM (SortOps α) β) (S : Finset (α → α → Bool)) :
+    (S.image fun le => run P le).card ≤ 2 ^ S.sup fun le => cost P le :=
+  card_image_run_le_two_pow_of_cost_le fun _ hle => Finset.le_sup hle
+
+section Sorting
+
+variable {n : ℕ}
+
+/-- The order on `Fin n` induced by a hidden permutation `σ`. -/
+def permLE (σ : Equiv.Perm (Fin n)) : Fin n → Fin n → Bool :=
+  fun i j => decide (σ i ≤ σ j)
+
+/-- The sorted output for the hidden order `permLE σ`. -/
+def permOutput (σ : Equiv.Perm (Fin n)) : List (Fin n) :=
+  List.ofFn σ.symm
+
+lemma permOutput_injective : Function.Injective (permOutput (n := n)) := fun _ _ h => by
+  simpa using congrArg Equiv.symm (Equiv.coe_fn_injective (List.ofFn_injective h))
+
+/-- Worst-case number of comparisons of `P` over all hidden permutation orders. -/
+def worstTime (P : FreeM (SortOps (Fin n)) (List (Fin n))) : ℕ :=
+  Finset.univ.sup fun σ : Equiv.Perm (Fin n) => cost P (permLE σ)
+
+/-- A program that sorts under every hidden permutation order distinguishes all `n !` of
+them. -/
+theorem factorial_le_two_pow_worstTime
+    (P : FreeM (SortOps (Fin n)) (List (Fin n)))
+    (hP : ∀ σ : Equiv.Perm (Fin n), run P (permLE σ) = permOutput σ) :
+    n ! ≤ 2 ^ worstTime P := by
+  classical
+  have h := card_image_run_le_two_pow P (Finset.univ.image permLE)
+  rw [Finset.sup_image, Finset.image_image] at h
+  rw [Function.comp_def, Function.comp_def,
+    Finset.image_congr (g := permOutput) fun σ _ => hP σ,
+    Finset.card_image_of_injective _ permOutput_injective, Finset.card_univ,
+    Fintype.card_perm, Fintype.card_fin] at h
+  exact h
+
+/-- Sorting under every hidden permutation order on `Fin n` takes at least `log₂ (n !)`
+comparisons in the worst case. -/
+theorem log_factorial_le_worstTime
+    (P : FreeM (SortOps (Fin n)) (List (Fin n)))
+    (hP : ∀ σ : Equiv.Perm (Fin n), run P (permLE σ) = permOutput σ) :
+    Nat.log 2 (n !) ≤ worstTime P :=
+  (Nat.log_mono_right (factorial_le_two_pow_worstTime P hP)).trans_eq
+    (Nat.log_pow Nat.one_lt_two _)
+
+/-- Explicit lower estimate for `log₂ (n !)`. -/
+lemma div_two_mul_log_le_log_factorial (n : ℕ) :
+    n / 2 * Nat.log 2 (n / 2) ≤ Nat.log 2 (n !) := by
+  set k := n / 2 with hk
+  by_cases hk0 : k = 0
+  · simp [hk0]
+  · have hkPow_le_factorial : k ^ k ≤ n ! :=
+      calc k ^ k ≤ k ^ (n - k) := Nat.pow_le_pow_right (Nat.pos_of_ne_zero hk0) (by omega)
+        _ ≤ k ! * k ^ (n - k) := Nat.le_mul_of_pos_left _ (Nat.factorial_pos k)
+        _ ≤ n ! := Nat.factorial_mul_pow_sub_le_factorial (hk ▸ Nat.div_le_self n 2)
+    refine le_trans (Nat.le_log_of_pow_le Nat.one_lt_two ?_)
+      (Nat.log_mono_right hkPow_le_factorial)
+    calc 2 ^ (k * Nat.log 2 k) = (2 ^ Nat.log 2 k) ^ k := by rw [Nat.mul_comm, Nat.pow_mul]
+      _ ≤ k ^ k := Nat.pow_le_pow_left (Nat.pow_log_le_self 2 hk0) k
+
+/-- The comparison sort lower bound in explicit `Ω(n log n)` form. -/
+theorem cmpSort_lower_bound
+    (P : FreeM (SortOps (Fin n)) (List (Fin n)))
+    (hP : ∀ σ : Equiv.Perm (Fin n), run P (permLE σ) = permOutput σ) :
+    n / 2 * Nat.log 2 (n / 2) ≤ worstTime P :=
+  (div_two_mul_log_le_log_factorial n).trans (log_factorial_le_worstTime P hP)
+
+end Sorting
+
+end Cslib.Algorithms