From e1ab2abe060292efe22c10dd0deba430f9835f01 Mon Sep 17 00:00:00 2001 From: Cameron Freer Date: Tue, 9 Jun 2026 23:42:23 +0000 Subject: [PATCH 1/6] style: rewrite 5 historical comments to describe current state Per the project style rule (describe what code is, not what it was): - AlphaIicCE.lean: the 'BorelSpace synthesis issue' note described a problem the proof below no longer has - L2Helpers.lean, DirectingMeasureIntegral.lean, AlphaIicEq.lean, CommonEnding.lean: drop 'previously'/'earlier drafts' phrasing --- Exchangeability/DeFinetti/CommonEnding.lean | 6 ++---- Exchangeability/DeFinetti/L2Helpers.lean | 4 ++-- .../DeFinetti/ViaL2/AlphaConvergence/AlphaIicEq.lean | 4 ++-- Exchangeability/DeFinetti/ViaL2/AlphaIicCE.lean | 7 +++---- .../DeFinetti/ViaL2/DirectingMeasureIntegral.lean | 4 ++-- 5 files changed, 11 insertions(+), 14 deletions(-) diff --git a/Exchangeability/DeFinetti/CommonEnding.lean b/Exchangeability/DeFinetti/CommonEnding.lean index ab07ac44..0d82deb9 100644 --- a/Exchangeability/DeFinetti/CommonEnding.lean +++ b/Exchangeability/DeFinetti/CommonEnding.lean @@ -425,8 +425,6 @@ theorem conditional_iid_from_directing_measure exact fidi_eq_avg_product X hX_meas ν hν_prob hν_meas m k B hB_meas (h_bridge (k := k) hk_strict.injective (B := B) hB_meas) --- *Monotone-class remark.* Earlier drafts included an explicit monotone-class lemma --- (`monotone_class_product_extension`) proving the π-λ step described above. The sole --- remaining use of that lemma is captured abstractly by the `h_bridge` hypothesis, so the --- sketch is retained only as commentary. +-- *Monotone-class remark.* The π-λ step described above is captured abstractly by the +-- `h_bridge` hypothesis, so no explicit monotone-class lemma is needed here. end Exchangeability.DeFinetti.CommonEnding diff --git a/Exchangeability/DeFinetti/L2Helpers.lean b/Exchangeability/DeFinetti/L2Helpers.lean index 43520262..384e39bc 100644 --- a/Exchangeability/DeFinetti/L2Helpers.lean +++ b/Exchangeability/DeFinetti/L2Helpers.lean @@ -24,8 +24,8 @@ All lemmas here are complete (no sorries) and compile cleanly. - Various arithmetic bounds for convergence rates (e.g. `sqrt_div_lt_half_eps_of_nat`) - Fin index reindexing lemmas for filtered sums -The contractability/covariance helpers that previously lived here have moved to the -canonical `Exchangeability.Contractable.{map_single, map_pair, comp}` API in +For contractability/covariance helpers, see the canonical +`Exchangeability.Contractable.{map_single, map_pair, comp}` API in `Exchangeability/Contractability.lean`. -/ diff --git a/Exchangeability/DeFinetti/ViaL2/AlphaConvergence/AlphaIicEq.lean b/Exchangeability/DeFinetti/ViaL2/AlphaConvergence/AlphaIicEq.lean index 5a5184ea..365841b8 100644 --- a/Exchangeability/DeFinetti/ViaL2/AlphaConvergence/AlphaIicEq.lean +++ b/Exchangeability/DeFinetti/ViaL2/AlphaConvergence/AlphaIicEq.lean @@ -34,8 +34,8 @@ variable {Ω α : Type*} [MeasurableSpace Ω] [MeasurableSpace α] /-- Clipping `b` to `[0,1]` can only bring it closer to anything already in `[0,1]`. -Internal helper: factors out the case analysis that previously appeared in -`alphaIic_ae_eq_alphaIicCE`. The cleanest route is via mathlib's +Internal helper for the case analysis in `alphaIic_ae_eq_alphaIicCE`. +The cleanest route is via mathlib's `LipschitzWith.projIcc`: `Set.projIcc 0 1` is `1`-Lipschitz, equals `max 0 (min 1 ·)` after coercion, and fixes any input already in `[0,1]`. -/ private lemma abs_sub_clip01_le_of_mem {a b : ℝ} (ha : a ∈ Set.Icc (0 : ℝ) 1) : diff --git a/Exchangeability/DeFinetti/ViaL2/AlphaIicCE.lean b/Exchangeability/DeFinetti/ViaL2/AlphaIicCE.lean index c07f3962..5afaa5d9 100644 --- a/Exchangeability/DeFinetti/ViaL2/AlphaIicCE.lean +++ b/Exchangeability/DeFinetti/ViaL2/AlphaIicCE.lean @@ -78,10 +78,9 @@ noncomputable def alphaIicCE /-- Measurability of alphaIicCE. -Note: Previously had BorelSpace typeclass instance resolution issues. -The conditional expectation `condExp μ (tailSigma X) f` is measurable by -`stronglyMeasurable_condExp.measurable`, but Lean can't synthesize the required -`BorelSpace` instance automatically. This should be straightforward to fix. -/ +The conditional expectation is strongly measurable with respect to the tail σ-algebra +(`stronglyMeasurable_condExp`), and `Measurable.mono` lifts this to the ambient +σ-algebra via `tailProcess_le_ambient`. -/ @[measurability, fun_prop] lemma alphaIicCE_measurable {μ : Measure Ω} [IsProbabilityMeasure μ] diff --git a/Exchangeability/DeFinetti/ViaL2/DirectingMeasureIntegral.lean b/Exchangeability/DeFinetti/ViaL2/DirectingMeasureIntegral.lean index 1d8fb7f9..3c04734c 100644 --- a/Exchangeability/DeFinetti/ViaL2/DirectingMeasureIntegral.lean +++ b/Exchangeability/DeFinetti/ViaL2/DirectingMeasureIntegral.lean @@ -44,8 +44,8 @@ variable {Ω α : Type*} [MeasurableSpace Ω] [MeasurableSpace α] The proof routes through the kernel-level bridge `directing_measure_ae_eq_condExpKernel_map` and mathlib's -`condExp_ae_eq_integral_condExpKernel`, replacing what was previously a hand-built -monotone-class extension from `Iic` indicators. +`condExp_ae_eq_integral_condExpKernel`, so no monotone-class extension from `Iic` +indicators is needed. The `[StandardBorelSpace Ω]` assumption comes from `condExpKernel`. -/ lemma directing_measure_integral_eq_condExp From 68c9e48bc26a3d277343246f1a521a0f6e383f68 Mon Sep 17 00:00:00 2001 From: Cameron Freer Date: Tue, 9 Jun 2026 23:42:23 +0000 Subject: [PATCH 2/6] refactor: consolidate ViaKoopman/LpCondExpHelpers + MartingaleExtras into shared Probability helpers Intentional API cleanup (breaking for external users of these names): - condExp_const_mul: removed; the two call sites in CesaroHelpers now use mathlib's condExp_smul directly - condExp_sum_finset: now a private lemma in its only consumer, CesaroHelpers.lean - integrable_of_bounded_measurable, eLpNorm_one_le_eLpNorm_two_toReal: moved to Probability/LpNormHelpers.lean (namespace MeasureTheory) - lintegral_fatou_ofReal_norm: moved from MartingaleExtras.lean into Probability/IntegrationHelpers.lean Module count 105 -> 103: regenerated import_graph_colored {svg,png} (same procedure as 722fba09, colors carried over per module) and refreshed file counts in README.md / home_page/index.md. --- .../DeFinetti/ViaKoopman/CesaroHelpers.lean | 29 +- .../ViaKoopman/LpCondExpHelpers.lean | 116 -- .../DeFinetti/ViaKoopman/README.md | 1 - .../Probability/IntegrationHelpers.lean | 26 + .../Probability/LpNormHelpers.lean | 39 + .../Martingale/Crossings/AntitoneLimit.lean | 4 +- .../Probability/MartingaleExtras.lean | 61 - README.md | 4 +- blueprint/web/import_graph_colored.png | Bin 272358 -> 308031 bytes blueprint/web/import_graph_colored.svg | 1160 ++++++++--------- home_page/index.md | 2 +- 11 files changed, 666 insertions(+), 776 deletions(-) delete mode 100644 Exchangeability/DeFinetti/ViaKoopman/LpCondExpHelpers.lean delete mode 100644 Exchangeability/Probability/MartingaleExtras.lean diff --git a/Exchangeability/DeFinetti/ViaKoopman/CesaroHelpers.lean b/Exchangeability/DeFinetti/ViaKoopman/CesaroHelpers.lean index 5c7add21..d1a83232 100644 --- a/Exchangeability/DeFinetti/ViaKoopman/CesaroHelpers.lean +++ b/Exchangeability/DeFinetti/ViaKoopman/CesaroHelpers.lean @@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Cameron Freer -/ import Exchangeability.DeFinetti.ViaKoopman.Infrastructure -import Exchangeability.DeFinetti.ViaKoopman.LpCondExpHelpers +import Exchangeability.Probability.LpNormHelpers /-! # Cesàro Helper Lemmas @@ -36,6 +36,25 @@ variable {α : Type*} [MeasurableSpace α] -- Short notation for shift-invariant σ-algebra (used throughout this file) local notation "mSI" => shiftInvariantSigma (α := α) +/-- Finite sum linearity of conditional expectation, stated with the sum applied +pointwise (`fun ω => ∑ i ∈ s, f i ω`) as it appears in the Cesàro averages below. +η-expands to mathlib's `condExp_finset_sum`. -/ +private lemma condExp_sum_finset + {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] + {m : MeasurableSpace Ω} (_hm : m ≤ mΩ) + {ι : Type*} (s : Finset ι) (f : ι → Ω → ℝ) + (hint : ∀ i ∈ s, Integrable (f i) μ) : + μ[(fun ω => s.sum (fun i => f i ω)) | m] + =ᵐ[μ] (fun ω => s.sum (fun i => μ[f i | m] ω)) := by + classical + have h_sum_eta : (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, f i := by + funext ω + simp only [Finset.sum_apply] + have h_ce_sum_eta : (fun ω => ∑ i ∈ s, μ[f i | m] ω) = ∑ i ∈ s, μ[f i | m] := by + funext ω + simp only [Finset.sum_apply] + rw [h_sum_eta, h_ce_sum_eta] + exact condExp_finset_sum hint m lemma condexp_precomp_iterate_eq {μ : Measure (Ω[α])} [IsProbabilityMeasure μ] @@ -321,8 +340,8 @@ lemma product_ce_constant_of_lag_const = (fun ω => (1 / ((n + 1) : ℝ)) * (Finset.range (n + 1)).sum (fun j => f (ω 0) * g (ω j))) := by funext ω; simp [A, Finset.mul_sum, mul_comm, mul_left_comm, mul_assoc] rw [this] - exact condExp_const_mul (shiftInvariantSigma_le (α := α)) - (1 / ((n + 1) : ℝ)) (fun ω => (Finset.range (n + 1)).sum (fun j => f (ω 0) * g (ω j))) + simpa [Pi.mul_apply, smul_eq_mul] using condExp_smul (μ := μ) + (1 / ((n + 1) : ℝ)) (fun ω => (Finset.range (n + 1)).sum (fun j => f (ω 0) * g (ω j))) mSI -- Push CE through the finite sum have h_sum : @@ -460,8 +479,8 @@ lemma product_ce_constant_of_lag_const_from_one = (fun ω => (1 / (n : ℝ)) * (Finset.range n).sum (fun j => f (ω 0) * g (ω (j + 1)))) := by funext ω; simp [A', Finset.mul_sum, mul_left_comm] rw [this] - exact condExp_const_mul (shiftInvariantSigma_le (α := α)) - (1 / (n : ℝ)) (fun ω => (Finset.range n).sum (fun j => f (ω 0) * g (ω (j + 1)))) + simpa [Pi.mul_apply, smul_eq_mul] using condExp_smul (μ := μ) + (1 / (n : ℝ)) (fun ω => (Finset.range n).sum (fun j => f (ω 0) * g (ω (j + 1)))) mSI -- Push CE through the finite sum have h_sum : diff --git a/Exchangeability/DeFinetti/ViaKoopman/LpCondExpHelpers.lean b/Exchangeability/DeFinetti/ViaKoopman/LpCondExpHelpers.lean deleted file mode 100644 index 0af2a937..00000000 --- a/Exchangeability/DeFinetti/ViaKoopman/LpCondExpHelpers.lean +++ /dev/null @@ -1,116 +0,0 @@ -/- -Copyright (c) 2025 Cameron Freer. All rights reserved. -Released under Apache 2.0 license as described in the file LICENSE. -Authors: Cameron Freer --/ - -import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic -import Mathlib.MeasureTheory.Function.LpSpace.Basic - -/-! -# Conditional Expectation and Lp Helper Lemmas - -This file contains utility lemmas for conditional expectation and Lp norm manipulation. - -## Main Definitions - -None - this file only contains helper lemmas. - -## Main Results - -* `condExp_const_mul`: Scalar linearity of conditional expectation -* `condExp_sum_finset`: Finite sum linearity of conditional expectation -* `integrable_of_bounded_measurable`: Bounded measurable functions are integrable on finite measure spaces -* `eLpNorm_one_le_eLpNorm_two_toReal`: L¹ norm bounded by L² norm on probability spaces --/ - -open MeasureTheory Filter -open scoped Topology - -noncomputable section - -/-! ### Lp norm placeholder -/ - -/-! ### Lp seminorm: use mathlib's `eLpNorm` -/ - -/-! ### Conditional expectation linearity helpers -/ - -/-- Scalar linearity of conditional expectation. -**Mathematical content**: CE[c·f| mSI] = c·CE[f| mSI] -**Mathlib source**: `MeasureTheory.condexp_smul` for scalar multiplication. -/ -lemma condExp_const_mul - {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] - {m : MeasurableSpace Ω} (_hm : m ≤ mΩ) - (c : ℝ) (f : Ω → ℝ) : - μ[(fun ω => c * f ω) | m] =ᵐ[μ] (fun ω => c * μ[f | m] ω) := by - -- `condExp_smul` in mathlib takes m as explicit positional parameter - simpa [Pi.mul_apply, smul_eq_mul] using - (MeasureTheory.condExp_smul c f m) - -/-- Finite sum linearity of conditional expectation. -**Mathematical content**: CE[Σᵢfᵢ| mSI] = ΣᵢCE[fᵢ| mSI] -**Mathlib source**: Direct application of `MeasureTheory.condExp_finset_sum`. -NOTE: Uses η-expansion to work around notation elaboration issues with `∑ i ∈ s, f i` vs `fun ω => ∑ i ∈ s, f i ω`. -/ -lemma condExp_sum_finset - {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] - {m : MeasurableSpace Ω} (_hm : m ≤ mΩ) - {ι : Type*} (s : Finset ι) (f : ι → Ω → ℝ) - (hint : ∀ i ∈ s, Integrable (f i) μ) : - μ[(fun ω => s.sum (fun i => f i ω)) | m] - =ᵐ[μ] (fun ω => s.sum (fun i => μ[f i | m] ω)) := by - classical - -- Rewrite using η-reduction: (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, f i - have h_sum_eta : (fun ω => ∑ i ∈ s, f i ω) = ∑ i ∈ s, f i := by - funext ω - simp only [Finset.sum_apply] - have h_ce_sum_eta : (fun ω => ∑ i ∈ s, μ[f i | m] ω) = ∑ i ∈ s, μ[f i | m] := by - funext ω - simp only [Finset.sum_apply] - -- Rewrite goal using η-reduction - rw [h_sum_eta, h_ce_sum_eta] - -- Apply condExp_finset_sum - exact condExp_finset_sum hint m - -/-- On a finite measure space, a bounded measurable real function is integrable. -/ -lemma integrable_of_bounded_measurable - {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] - {f : Ω → ℝ} (hf_meas : Measurable f) (C : ℝ) (hf_bd : ∀ ω, |f ω| ≤ C) : - Integrable f μ := - ⟨hf_meas.aestronglyMeasurable, HasFiniteIntegral.of_bounded (by - filter_upwards with ω; simpa [Real.norm_eq_abs] using hf_bd ω)⟩ - -/-- On a probability space, `‖f‖₁ ≤ ‖f‖₂`. Version with real integral on the left. -We assume `MemLp f 2 μ` so the right-hand side is finite; this matches all uses below -where the function is bounded (hence in L²). - -**Proof strategy** (from user's specification): -- Use `snorm_mono_exponent` or `memℒp_one_of_memℒp_two` to get `MemLp f 1 μ` from `MemLp f 2 μ` -- Show both `eLpNorm f 1 μ` and `eLpNorm f 2 μ` are finite -- Apply exponent monotonicity: `eLpNorm f 1 μ ≤ eLpNorm f 2 μ` on probability spaces -- Convert `∫|f|` to `(eLpNorm f 1 μ).toReal` and apply `ENNReal.toReal_le_toReal` --/ -lemma eLpNorm_one_le_eLpNorm_two_toReal - {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] - (f : Ω → ℝ) (hL1 : Integrable f μ) (hL2 : MemLp f 2 μ) : - (∫ ω, |f ω| ∂μ) ≤ (eLpNorm f 2 μ).toReal := by - -- Step 1: Connect ∫|f| to eLpNorm f 1 μ using norm - have h_eq : ENNReal.ofReal (∫ ω, |f ω| ∂μ) = eLpNorm f 1 μ := by - have h_norm : ∫ ω, |f ω| ∂μ = ∫ ω, ‖f ω‖ ∂μ := integral_congr_ae (ae_of_all μ (fun ω => (Real.norm_eq_abs (f ω)).symm)) - rw [h_norm, ofReal_integral_norm_eq_lintegral_enorm hL1] - exact eLpNorm_one_eq_lintegral_enorm.symm - - -- Step 2: eLpNorm f 1 μ ≤ eLpNorm f 2 μ on probability spaces - have h_mono : eLpNorm f 1 μ ≤ eLpNorm f 2 μ := by - have h_ae : AEStronglyMeasurable f μ := hL1.aestronglyMeasurable - refine eLpNorm_le_eLpNorm_of_exponent_le ?_ h_ae - norm_num - - -- Step 3: Convert to toReal inequality - have h_fin : eLpNorm f 2 μ ≠ ⊤ := hL2.eLpNorm_ne_top - have h_nonneg : 0 ≤ ∫ ω, |f ω| ∂μ := integral_nonneg (fun ω => abs_nonneg _) - calc (∫ ω, |f ω| ∂μ) - = (ENNReal.ofReal (∫ ω, |f ω| ∂μ)).toReal := by - rw [ENNReal.toReal_ofReal h_nonneg] - _ = (eLpNorm f 1 μ).toReal := by rw [h_eq] - _ ≤ (eLpNorm f 2 μ).toReal := ENNReal.toReal_mono h_fin h_mono - diff --git a/Exchangeability/DeFinetti/ViaKoopman/README.md b/Exchangeability/DeFinetti/ViaKoopman/README.md index bdd82c2a..d655ddc3 100644 --- a/Exchangeability/DeFinetti/ViaKoopman/README.md +++ b/Exchangeability/DeFinetti/ViaKoopman/README.md @@ -32,7 +32,6 @@ This directory contains Kallenberg's "first proof" of de Finetti's theorem, usin | `BlockInjection.lean` | Block injection lemmas | | `KernelBridge.lean` | Bridge lemma for kernel measures | | `CylinderFunctions.lean` | Cylinder function definitions | -| `LpCondExpHelpers.lean` | Lᵖ conditional expectation helpers | | `DirectingKernel.lean` | Directing measure from CE kernel | | `Quantization.lean` | Quantization machinery | | `Infrastructure.lean` | Basic definitions | diff --git a/Exchangeability/Probability/IntegrationHelpers.lean b/Exchangeability/Probability/IntegrationHelpers.lean index 21197e01..0ddd0daf 100644 --- a/Exchangeability/Probability/IntegrationHelpers.lean +++ b/Exchangeability/Probability/IntegrationHelpers.lean @@ -193,4 +193,30 @@ lemma integral_pushforward_sq_diff rw [integral_map hf.aemeasurable] exact (continuous_id.sub continuous_const).pow 2 |>.aestronglyMeasurable +/-! ## Fatou-Type Lemmas -/ + +/-- Fatou's lemma on `ENNReal.ofReal ∘ ‖·‖` along an a.e. pointwise limit. + +If `u n x → g x` a.e., then `∫⁻ ‖g‖ ≤ liminf (∫⁻ ‖u n‖)`. -/ +lemma lintegral_fatou_ofReal_norm + {α β : Type*} [MeasurableSpace α] {μ : Measure α} + [MeasurableSpace β] [NormedAddCommGroup β] [BorelSpace β] + {u : ℕ → α → β} {g : α → β} + (hae : ∀ᵐ x ∂μ, Tendsto (fun n => u n x) atTop (nhds (g x))) + (hu_meas : ∀ n, AEMeasurable (fun x => ENNReal.ofReal ‖u n x‖) μ) + (_hg_meas : AEMeasurable (fun x => ENNReal.ofReal ‖g x‖) μ) : + ∫⁻ x, ENNReal.ofReal ‖g x‖ ∂μ + ≤ liminf (fun n => ∫⁻ x, ENNReal.ofReal ‖u n x‖ ∂μ) atTop := by + have hae_ofReal : + ∀ᵐ x ∂μ, + Tendsto (fun n => ENNReal.ofReal ‖u n x‖) atTop + (nhds (ENNReal.ofReal ‖g x‖)) := + hae.mono (fun x hx => + ((ENNReal.continuous_ofReal.comp continuous_norm).tendsto _).comp hx) + calc ∫⁻ x, ENNReal.ofReal ‖g x‖ ∂μ + = ∫⁻ x, liminf (fun n => ENNReal.ofReal ‖u n x‖) atTop ∂μ := + lintegral_congr_ae (hae_ofReal.mono fun x hx => hx.liminf_eq.symm) + _ ≤ liminf (fun n => ∫⁻ x, ENNReal.ofReal ‖u n x‖ ∂μ) atTop := + lintegral_liminf_le' hu_meas + end Exchangeability.Probability.IntegrationHelpers diff --git a/Exchangeability/Probability/LpNormHelpers.lean b/Exchangeability/Probability/LpNormHelpers.lean index 0c5e8a59..293b3201 100644 --- a/Exchangeability/Probability/LpNormHelpers.lean +++ b/Exchangeability/Probability/LpNormHelpers.lean @@ -5,6 +5,7 @@ Authors: Cameron Freer -/ import Mathlib.MeasureTheory.Function.L2Space import Mathlib.MeasureTheory.Function.LpSeminorm.Basic +import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp import Mathlib.Analysis.SpecialFunctions.Pow.Real /-! @@ -157,5 +158,43 @@ lemma memLp_two_of_bounded filter_upwards with ω exact (Real.norm_eq_abs _).le.trans (hf_bdd ω) +/-- On a finite measure space, a bounded measurable real function is integrable. + +Unlike mathlib's `Integrable.of_bound`, this takes `Measurable` and a pointwise +absolute-value bound, the shape produced by the block-average constructions. -/ +lemma integrable_of_bounded_measurable + [IsFiniteMeasure μ] + {f : Ω → ℝ} (hf_meas : Measurable f) (C : ℝ) (hf_bd : ∀ ω, |f ω| ≤ C) : + Integrable f μ := + ⟨hf_meas.aestronglyMeasurable, HasFiniteIntegral.of_bounded (by + filter_upwards with ω; simpa [Real.norm_eq_abs] using hf_bd ω)⟩ + +/-- On a probability space, `‖f‖₁ ≤ ‖f‖₂`. Version with real integral on the left. +We assume `MemLp f 2 μ` so the right-hand side is finite; this matches the uses +where the function is bounded (hence in L²). -/ +lemma eLpNorm_one_le_eLpNorm_two_toReal + [IsProbabilityMeasure μ] + (f : Ω → ℝ) (hL1 : Integrable f μ) (hL2 : MemLp f 2 μ) : + (∫ ω, |f ω| ∂μ) ≤ (eLpNorm f 2 μ).toReal := by + -- Step 1: Connect ∫|f| to eLpNorm f 1 μ using norm + have h_eq : ENNReal.ofReal (∫ ω, |f ω| ∂μ) = eLpNorm f 1 μ := by + have h_norm : ∫ ω, |f ω| ∂μ = ∫ ω, ‖f ω‖ ∂μ := + integral_congr_ae (ae_of_all μ (fun ω => (Real.norm_eq_abs (f ω)).symm)) + rw [h_norm, ofReal_integral_norm_eq_lintegral_enorm hL1] + exact eLpNorm_one_eq_lintegral_enorm.symm + -- Step 2: eLpNorm f 1 μ ≤ eLpNorm f 2 μ on probability spaces + have h_mono : eLpNorm f 1 μ ≤ eLpNorm f 2 μ := by + have h_ae : AEStronglyMeasurable f μ := hL1.aestronglyMeasurable + refine eLpNorm_le_eLpNorm_of_exponent_le ?_ h_ae + norm_num + -- Step 3: Convert to toReal inequality + have h_fin : eLpNorm f 2 μ ≠ ⊤ := hL2.eLpNorm_ne_top + have h_nonneg : 0 ≤ ∫ ω, |f ω| ∂μ := integral_nonneg (fun ω => abs_nonneg _) + calc (∫ ω, |f ω| ∂μ) + = (ENNReal.ofReal (∫ ω, |f ω| ∂μ)).toReal := by + rw [ENNReal.toReal_ofReal h_nonneg] + _ = (eLpNorm f 1 μ).toReal := by rw [h_eq] + _ ≤ (eLpNorm f 2 μ).toReal := ENNReal.toReal_mono h_fin h_mono + end MeasureTheory diff --git a/Exchangeability/Probability/Martingale/Crossings/AntitoneLimit.lean b/Exchangeability/Probability/Martingale/Crossings/AntitoneLimit.lean index 3f2d7931..dbf8e268 100644 --- a/Exchangeability/Probability/Martingale/Crossings/AntitoneLimit.lean +++ b/Exchangeability/Probability/Martingale/Crossings/AntitoneLimit.lean @@ -5,7 +5,7 @@ Authors: Cameron Freer -/ import Mathlib.Probability.Martingale.Convergence -import Exchangeability.Probability.MartingaleExtras +import Exchangeability.Probability.IntegrationHelpers import Exchangeability.Probability.SigmaAlgebraHelpers import Exchangeability.Probability.Martingale.Crossings.Bounds @@ -238,7 +238,7 @@ lemma condExp_exists_ae_limit_antitone calc ∫⁻ ω, ENNReal.ofReal ‖Xlim ω‖ ∂μ ≤ liminf (fun n => ∫⁻ ω, ENNReal.ofReal ‖μ[f | 𝔽 n] ω‖ ∂μ) atTop := - lintegral_fatou_ofReal_norm h_ae_tendsto hmeas_n hmeas_lim + IntegrationHelpers.lintegral_fatou_ofReal_norm h_ae_tendsto hmeas_n hmeas_lim _ ≤ ↑R := by rw [liminf_le_iff] intro b hb diff --git a/Exchangeability/Probability/MartingaleExtras.lean b/Exchangeability/Probability/MartingaleExtras.lean deleted file mode 100644 index 79ab0c12..00000000 --- a/Exchangeability/Probability/MartingaleExtras.lean +++ /dev/null @@ -1,61 +0,0 @@ -/- -Copyright (c) 2025 Cameron Freer. All rights reserved. -Released under Apache 2.0 license as described in the file LICENSE. -Authors: Cameron Freer --/ -import Mathlib.Probability.Martingale.Basic -import Mathlib.MeasureTheory.Function.LpSeminorm.Basic -import Mathlib.MeasureTheory.Integral.Bochner.Basic -import Mathlib.Probability.Kernel.CondDistrib -import Mathlib.Probability.Kernel.Composition.Comp -import Mathlib.MeasureTheory.MeasurableSpace.Prod -import Mathlib.MeasureTheory.Function.FactorsThrough - -/-! -# Martingale Helper Lemmas - -## Contents - -* `lintegral_fatou_ofReal_norm`: Fatou's lemma for `ENNReal.ofReal ∘ ‖·‖` - -## References - -* Durrett, *Probability: Theory and Examples* (2019), Section 5.5 -* Williams, *Probability with Martingales* (1991) --/ - -noncomputable section -open scoped MeasureTheory ProbabilityTheory Topology -open MeasureTheory Filter Set Function - -namespace Exchangeability.Probability - -variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} - -/-! ## Fatou-Type Lemmas -/ - -/-- Fatou's lemma on `ENNReal.ofReal ∘ ‖·‖` along an a.e. pointwise limit. - -If `u n x → g x` a.e., then `∫⁻ ‖g‖ ≤ liminf (∫⁻ ‖u n‖)`. -/ -lemma lintegral_fatou_ofReal_norm - {α β : Type*} [MeasurableSpace α] {μ : Measure α} - [MeasurableSpace β] [NormedAddCommGroup β] [BorelSpace β] - {u : ℕ → α → β} {g : α → β} - (hae : ∀ᵐ x ∂μ, Tendsto (fun n => u n x) atTop (nhds (g x))) - (hu_meas : ∀ n, AEMeasurable (fun x => ENNReal.ofReal ‖u n x‖) μ) - (_hg_meas : AEMeasurable (fun x => ENNReal.ofReal ‖g x‖) μ) : - ∫⁻ x, ENNReal.ofReal ‖g x‖ ∂μ - ≤ liminf (fun n => ∫⁻ x, ENNReal.ofReal ‖u n x‖ ∂μ) atTop := by - have hae_ofReal : - ∀ᵐ x ∂μ, - Tendsto (fun n => ENNReal.ofReal ‖u n x‖) atTop - (nhds (ENNReal.ofReal ‖g x‖)) := - hae.mono (fun x hx => - ((ENNReal.continuous_ofReal.comp continuous_norm).tendsto _).comp hx) - calc ∫⁻ x, ENNReal.ofReal ‖g x‖ ∂μ - = ∫⁻ x, liminf (fun n => ENNReal.ofReal ‖u n x‖) atTop ∂μ := - lintegral_congr_ae (hae_ofReal.mono fun x hx => hx.liminf_eq.symm) - _ ≤ liminf (fun n => ∫⁻ x, ENNReal.ofReal ‖u n x‖ ∂μ) atTop := - lintegral_liminf_le' hu_meas - -end Exchangeability.Probability diff --git a/README.md b/README.md index 3baebfcb..6ed0ee91 100644 --- a/README.md +++ b/README.md @@ -27,7 +27,7 @@ We implement **all three proofs** from Kallenberg (2005) of the key implication 3. **Koopman Approach** - Kallenberg's "first proof" - Mean Ergodic Theorem - Deep connection to dynamical systems and ergodic theory - - [`Exchangeability/DeFinetti/ViaKoopman/`](Exchangeability/DeFinetti/ViaKoopman/) (16 files) + - [`Exchangeability/DeFinetti/ViaKoopman/`](Exchangeability/DeFinetti/ViaKoopman/) (15 files) ### Import Graph @@ -94,7 +94,7 @@ Exchangeability/ │ ├── Theorem.lean # Public API (exports ViaMartingale) │ ├── ViaMartingale/ # Martingale proof (12 files) │ ├── ViaL2/ # L² proof (24 files) -│ ├── ViaKoopman/ # Ergodic proof (16 files) +│ ├── ViaKoopman/ # Ergodic proof (15 files) │ ├── CommonEnding.lean # Shared final step │ └── L2Helpers.lean # 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Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L2 - -Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L2 + +Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L2 Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L1 - -Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L1 + +Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L1 Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L2->Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L1 - - + + Exchangeability.DeFinetti.ViaL2.CesaroConvergence.Cauchy - -Exchangeability.DeFinetti.ViaL2.CesaroConvergence.Cauchy + +Exchangeability.DeFinetti.ViaL2.CesaroConvergence.Cauchy Exchangeability.DeFinetti.ViaL2.CesaroConvergence.Cauchy->Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L2 - - + + Exchangeability.DeFinetti.ViaL2.BlockAverages - -Exchangeability.DeFinetti.ViaL2.BlockAverages + +Exchangeability.DeFinetti.ViaL2.BlockAverages Exchangeability.DeFinetti.ViaL2.BlockAverages->Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L2 - - + + - + Exchangeability.DeFinetti.ViaL2.MainConvergence - -Exchangeability.DeFinetti.ViaL2.MainConvergence + +Exchangeability.DeFinetti.ViaL2.MainConvergence Exchangeability.DeFinetti.ViaL2.BlockAverages->Exchangeability.DeFinetti.ViaL2.MainConvergence - - + + Exchangeability.Tail.ShiftInvariantMeasure - -Exchangeability.Tail.ShiftInvariantMeasure + +Exchangeability.Tail.ShiftInvariantMeasure Exchangeability.Tail.ShiftInvariantMeasure->Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L2 - - + + Exchangeability.Tail.CondExpShiftInvariance - -Exchangeability.Tail.CondExpShiftInvariance + +Exchangeability.Tail.CondExpShiftInvariance Exchangeability.Tail.ShiftInvariantMeasure->Exchangeability.Tail.CondExpShiftInvariance - - + + Exchangeability.Probability.SigmaAlgebraHelpers - -Exchangeability.Probability.SigmaAlgebraHelpers + +Exchangeability.Probability.SigmaAlgebraHelpers Exchangeability.Probability.SigmaAlgebraHelpers->Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L2 - - + + Exchangeability.Probability.Martingale.Crossings.AntitoneLimit - -Exchangeability.Probability.Martingale.Crossings.AntitoneLimit + +Exchangeability.Probability.Martingale.Crossings.AntitoneLimit Exchangeability.Probability.SigmaAlgebraHelpers->Exchangeability.Probability.Martingale.Crossings.AntitoneLimit - - + + Exchangeability.DeFinetti.ViaKoopman - -Exchangeability.DeFinetti.ViaKoopman + +Exchangeability.DeFinetti.ViaKoopman - + Exchangeability.DeFinetti.TheoremViaKoopman - -Exchangeability.DeFinetti.TheoremViaKoopman + +Exchangeability.DeFinetti.TheoremViaKoopman Exchangeability.DeFinetti.ViaKoopman->Exchangeability.DeFinetti.TheoremViaKoopman - - + + Exchangeability.DeFinetti.ViaKoopman.ContractableFactorization - -Exchangeability.DeFinetti.ViaKoopman.ContractableFactorization + +Exchangeability.DeFinetti.ViaKoopman.ContractableFactorization Exchangeability.DeFinetti.ViaKoopman.ContractableFactorization->Exchangeability.DeFinetti.ViaKoopman - - + + Exchangeability.PathSpace.Shift - -Exchangeability.PathSpace.Shift + +Exchangeability.PathSpace.Shift Exchangeability.PathSpace.Shift->Exchangeability.Tail.ShiftInvariantMeasure - - + + Exchangeability.Ergodic.ShiftInvariantSigma - -Exchangeability.Ergodic.ShiftInvariantSigma + +Exchangeability.Ergodic.ShiftInvariantSigma Exchangeability.PathSpace.Shift->Exchangeability.Ergodic.ShiftInvariantSigma - - + + Exchangeability.DeFinetti.CommonEnding - -Exchangeability.DeFinetti.CommonEnding + +Exchangeability.DeFinetti.CommonEnding Exchangeability.PathSpace.Shift->Exchangeability.DeFinetti.CommonEnding - - + + Exchangeability.Ergodic.KoopmanMeanErgodic - -Exchangeability.Ergodic.KoopmanMeanErgodic + +Exchangeability.Ergodic.KoopmanMeanErgodic Exchangeability.PathSpace.Shift->Exchangeability.Ergodic.KoopmanMeanErgodic - - + + Exchangeability.PathSpace.CylinderHelpers - -Exchangeability.PathSpace.CylinderHelpers + +Exchangeability.PathSpace.CylinderHelpers Exchangeability.PathSpace.Shift->Exchangeability.PathSpace.CylinderHelpers - - + + Exchangeability.DeFinetti.ViaKoopman.CylinderFunctions - -Exchangeability.DeFinetti.ViaKoopman.CylinderFunctions + +Exchangeability.DeFinetti.ViaKoopman.CylinderFunctions Exchangeability.PathSpace.Shift->Exchangeability.DeFinetti.ViaKoopman.CylinderFunctions - - + + - + Exchangeability.Bridge.CesaroToCondExp - -Exchangeability.Bridge.CesaroToCondExp + +Exchangeability.Bridge.CesaroToCondExp Exchangeability.PathSpace.Shift->Exchangeability.Bridge.CesaroToCondExp - - + + Exchangeability.Util.ProductBounds - -Exchangeability.Util.ProductBounds + +Exchangeability.Util.ProductBounds Exchangeability.DeFinetti.ViaKoopman.BlockAverage - -Exchangeability.DeFinetti.ViaKoopman.BlockAverage + +Exchangeability.DeFinetti.ViaKoopman.BlockAverage Exchangeability.Util.ProductBounds->Exchangeability.DeFinetti.ViaKoopman.BlockAverage - - + + Exchangeability.DeFinetti.ViaL2.MoreL2Helpers - -Exchangeability.DeFinetti.ViaL2.MoreL2Helpers + +Exchangeability.DeFinetti.ViaL2.MoreL2Helpers Exchangeability.Util.ProductBounds->Exchangeability.DeFinetti.ViaL2.MoreL2Helpers - - + + Exchangeability.DeFinetti.ViaKoopman.BlockAverage->Exchangeability.DeFinetti.ViaKoopman.ContractableFactorization - - + + Exchangeability.Probability.Martingale - -Exchangeability.Probability.Martingale + +Exchangeability.Probability.Martingale - + Exchangeability.DeFinetti.ViaMartingale.Factorization - -Exchangeability.DeFinetti.ViaMartingale.Factorization + +Exchangeability.DeFinetti.ViaMartingale.Factorization Exchangeability.Probability.Martingale->Exchangeability.DeFinetti.ViaMartingale.Factorization - - + + Exchangeability.Probability.Martingale.Convergence - -Exchangeability.Probability.Martingale.Convergence + +Exchangeability.Probability.Martingale.Convergence Exchangeability.Probability.Martingale.Convergence->Exchangeability.Probability.Martingale - - + + Exchangeability.DeFinetti.ViaMartingale.KallenbergChain - -Exchangeability.DeFinetti.ViaMartingale.KallenbergChain + +Exchangeability.DeFinetti.ViaMartingale.KallenbergChain Exchangeability.Probability.Martingale.Convergence->Exchangeability.DeFinetti.ViaMartingale.KallenbergChain - - + + Exchangeability.Probability.CondIndep.Indicator - -Exchangeability.Probability.CondIndep.Indicator + +Exchangeability.Probability.CondIndep.Indicator Exchangeability.Probability.CondIndep.Bounded.Projection - -Exchangeability.Probability.CondIndep.Bounded.Projection + +Exchangeability.Probability.CondIndep.Bounded.Projection Exchangeability.Probability.CondIndep.Indicator->Exchangeability.Probability.CondIndep.Bounded.Projection - - + + Exchangeability.Probability.CondIndep.Basic - -Exchangeability.Probability.CondIndep.Basic + +Exchangeability.Probability.CondIndep.Basic Exchangeability.Probability.CondIndep.Basic->Exchangeability.Probability.CondIndep.Indicator - - + + Exchangeability.Probability.TripleLawDropInfo.PairLawHelpers - -Exchangeability.Probability.TripleLawDropInfo.PairLawHelpers + +Exchangeability.Probability.TripleLawDropInfo.PairLawHelpers - + Exchangeability.Probability.TripleLawDropInfo.DropInfo - -Exchangeability.Probability.TripleLawDropInfo.DropInfo + +Exchangeability.Probability.TripleLawDropInfo.DropInfo Exchangeability.Probability.TripleLawDropInfo.PairLawHelpers->Exchangeability.Probability.TripleLawDropInfo.DropInfo - - + + Exchangeability.DeFinetti.ViaKoopman.BlockInjection - -Exchangeability.DeFinetti.ViaKoopman.BlockInjection + +Exchangeability.DeFinetti.ViaKoopman.BlockInjection Exchangeability.Ergodic.ShiftInvariantSigma->Exchangeability.DeFinetti.ViaKoopman.BlockInjection - - + + Exchangeability.DeFinetti.ViaKoopman.DirectingKernel - -Exchangeability.DeFinetti.ViaKoopman.DirectingKernel + +Exchangeability.DeFinetti.ViaKoopman.DirectingKernel Exchangeability.Ergodic.ShiftInvariantSigma->Exchangeability.DeFinetti.ViaKoopman.DirectingKernel - - + + - + Exchangeability.Ergodic.ShiftInvariantRepresentatives - -Exchangeability.Ergodic.ShiftInvariantRepresentatives + +Exchangeability.Ergodic.ShiftInvariantRepresentatives Exchangeability.Ergodic.ShiftInvariantSigma->Exchangeability.Ergodic.ShiftInvariantRepresentatives - - + + Exchangeability.DeFinetti.ViaMartingale.CondExpConvergence - -Exchangeability.DeFinetti.ViaMartingale.CondExpConvergence + +Exchangeability.DeFinetti.ViaMartingale.CondExpConvergence Exchangeability.DeFinetti.ViaMartingale.DirectingMeasure - -Exchangeability.DeFinetti.ViaMartingale.DirectingMeasure + +Exchangeability.DeFinetti.ViaMartingale.DirectingMeasure Exchangeability.DeFinetti.ViaMartingale.CondExpConvergence->Exchangeability.DeFinetti.ViaMartingale.DirectingMeasure - - + + Exchangeability.DeFinetti.ViaMartingale.CondExpConvergence->Exchangeability.DeFinetti.ViaMartingale.Factorization - - + + Exchangeability.DeFinetti.ViaMartingale.FutureFiltration - -Exchangeability.DeFinetti.ViaMartingale.FutureFiltration + +Exchangeability.DeFinetti.ViaMartingale.FutureFiltration Exchangeability.DeFinetti.ViaMartingale.FutureFiltration->Exchangeability.DeFinetti.ViaMartingale.CondExpConvergence - - + + Exchangeability.DeFinetti.ViaMartingale.FutureRectangles - -Exchangeability.DeFinetti.ViaMartingale.FutureRectangles + +Exchangeability.DeFinetti.ViaMartingale.FutureRectangles Exchangeability.DeFinetti.ViaMartingale.FutureRectangles->Exchangeability.DeFinetti.ViaMartingale.CondExpConvergence - - + + Exchangeability.DeFinetti.ViaMartingale.KallenbergChain->Exchangeability.DeFinetti.ViaMartingale.CondExpConvergence - - + + Exchangeability.DeFinetti.ViaKoopman.BlockInjection->Exchangeability.DeFinetti.ViaKoopman.BlockAverage - - + + Exchangeability.DeFinetti.ViaKoopman.CesaroConvergence - -Exchangeability.DeFinetti.ViaKoopman.CesaroConvergence + +Exchangeability.DeFinetti.ViaKoopman.CesaroConvergence Exchangeability.DeFinetti.ViaKoopman.CesaroConvergence->Exchangeability.DeFinetti.ViaKoopman.BlockAverage - - + + Exchangeability.DeFinetti.ViaKoopman.DirectingKernel->Exchangeability.DeFinetti.ViaKoopman.BlockAverage - - + + Exchangeability.DeFinetti.BridgeProperty - -Exchangeability.DeFinetti.BridgeProperty + +Exchangeability.DeFinetti.BridgeProperty Exchangeability.DeFinetti.BridgeProperty->Exchangeability.DeFinetti.ViaL2.MoreL2Helpers - - + + Exchangeability.DeFinetti.ViaMartingale.FiniteProduct - -Exchangeability.DeFinetti.ViaMartingale.FiniteProduct + +Exchangeability.DeFinetti.ViaMartingale.FiniteProduct Exchangeability.DeFinetti.ViaMartingale.FiniteProduct->Exchangeability.DeFinetti.BridgeProperty - - + + Exchangeability.DeFinetti.ViaMartingale - -Exchangeability.DeFinetti.ViaMartingale + +Exchangeability.DeFinetti.ViaMartingale Exchangeability.DeFinetti.ViaMartingale.FiniteProduct->Exchangeability.DeFinetti.ViaMartingale - - + + Exchangeability.Util.StrictMono - -Exchangeability.Util.StrictMono + +Exchangeability.Util.StrictMono - + Exchangeability.Contractability - -Exchangeability.Contractability + +Exchangeability.Contractability Exchangeability.Util.StrictMono->Exchangeability.Contractability - - + + Exchangeability.Probability.CondIndep.Bounded - -Exchangeability.Probability.CondIndep.Bounded + +Exchangeability.Probability.CondIndep.Bounded Exchangeability.Probability.CondIndep.Bounded.Projection->Exchangeability.Probability.CondIndep.Bounded - - + + Exchangeability.Tail.TailSigma - -Exchangeability.Tail.TailSigma + +Exchangeability.Tail.TailSigma Exchangeability.Tail.TailSigma->Exchangeability.Tail.ShiftInvariantMeasure - - + + Exchangeability.DeFinetti.ViaMartingale.ShiftOperations - -Exchangeability.DeFinetti.ViaMartingale.ShiftOperations + +Exchangeability.DeFinetti.ViaMartingale.ShiftOperations Exchangeability.Tail.TailSigma->Exchangeability.DeFinetti.ViaMartingale.ShiftOperations - - + + Exchangeability.Tail.TailSigma->Exchangeability.DeFinetti.CommonEnding - - + + Exchangeability.Tail.TailSigma->Exchangeability.DeFinetti.ViaL2.MainConvergence - - + + Exchangeability.Core - -Exchangeability.Core + +Exchangeability.Core Exchangeability.Core->Exchangeability.Tail.ShiftInvariantMeasure - - + + Exchangeability.Core->Exchangeability.DeFinetti.ViaMartingale.KallenbergChain - - + + Exchangeability.DeFinetti.ViaMartingale.PairLawEquality - -Exchangeability.DeFinetti.ViaMartingale.PairLawEquality + +Exchangeability.DeFinetti.ViaMartingale.PairLawEquality Exchangeability.Core->Exchangeability.DeFinetti.ViaMartingale.PairLawEquality - - + + Exchangeability.DeFinetti.ViaKoopman.InfraCore - -Exchangeability.DeFinetti.ViaKoopman.InfraCore + +Exchangeability.DeFinetti.ViaKoopman.InfraCore Exchangeability.Core->Exchangeability.DeFinetti.ViaKoopman.InfraCore - - + + Exchangeability.Core->Exchangeability.Bridge.CesaroToCondExp - - + + Exchangeability.Probability.Martingale.Crossings.Pathwise - -Exchangeability.Probability.Martingale.Crossings.Pathwise + +Exchangeability.Probability.Martingale.Crossings.Pathwise Exchangeability.Probability.Martingale.Crossings.Bounds - -Exchangeability.Probability.Martingale.Crossings.Bounds + +Exchangeability.Probability.Martingale.Crossings.Bounds Exchangeability.Probability.Martingale.Crossings.Pathwise->Exchangeability.Probability.Martingale.Crossings.Bounds - - + + Exchangeability.Probability.TimeReversalCrossing - -Exchangeability.Probability.TimeReversalCrossing + +Exchangeability.Probability.TimeReversalCrossing Exchangeability.Probability.TimeReversalCrossing->Exchangeability.Probability.Martingale.Crossings.Pathwise - - + + - + Exchangeability.Probability.Martingale.Crossings - -Exchangeability.Probability.Martingale.Crossings + +Exchangeability.Probability.Martingale.Crossings Exchangeability.Probability.Martingale.Crossings.AntitoneLimit->Exchangeability.Probability.Martingale.Crossings - - + + - - + + -Exchangeability.Probability.MartingaleExtras - -Exchangeability.Probability.MartingaleExtras +Exchangeability.Probability.IntegrationHelpers + +Exchangeability.Probability.IntegrationHelpers - + -Exchangeability.Probability.MartingaleExtras->Exchangeability.Probability.Martingale.Crossings.AntitoneLimit - - +Exchangeability.Probability.IntegrationHelpers->Exchangeability.Probability.Martingale.Crossings.AntitoneLimit + + + + + + +Exchangeability.Probability.CenteredVariables + +Exchangeability.Probability.CenteredVariables + + + + +Exchangeability.Probability.IntegrationHelpers->Exchangeability.Probability.CenteredVariables + + + + + + +Exchangeability.Probability.CondExp + +Exchangeability.Probability.CondExp + + + + +Exchangeability.Probability.IntegrationHelpers->Exchangeability.Probability.CondExp + + Exchangeability.Probability.Martingale.Crossings.Bounds->Exchangeability.Probability.Martingale.Crossings.AntitoneLimit - - + + Exchangeability.DeFinetti.ViaL2.BlockAvgDef - -Exchangeability.DeFinetti.ViaL2.BlockAvgDef + +Exchangeability.DeFinetti.ViaL2.BlockAvgDef Exchangeability.DeFinetti.ViaL2.BlockAvgDef->Exchangeability.DeFinetti.ViaL2.CesaroConvergence.Cauchy - - + + - + Exchangeability.Probability.CondIndep - -Exchangeability.Probability.CondIndep + +Exchangeability.Probability.CondIndep Exchangeability.Probability.CondIndep.Bounded->Exchangeability.Probability.CondIndep - - + + Exchangeability.DeFinetti.ViaMartingale.PairLawEquality->Exchangeability.DeFinetti.ViaMartingale.Factorization - - + + Exchangeability.Probability.TripleLawDropInfo - -Exchangeability.Probability.TripleLawDropInfo + +Exchangeability.Probability.TripleLawDropInfo Exchangeability.Probability.TripleLawDropInfo->Exchangeability.DeFinetti.ViaMartingale.PairLawEquality - - + + Exchangeability.DeFinetti.ViaMartingale.ShiftOperations->Exchangeability.DeFinetti.ViaMartingale.FutureRectangles - - + + Exchangeability.DeFinetti.ViaMartingale.ShiftOperations->Exchangeability.DeFinetti.ViaMartingale.PairLawEquality - - + + - + Exchangeability.DeFinetti.ViaMartingale.RevFiltration - -Exchangeability.DeFinetti.ViaMartingale.RevFiltration + +Exchangeability.DeFinetti.ViaMartingale.RevFiltration Exchangeability.DeFinetti.ViaMartingale.ShiftOperations->Exchangeability.DeFinetti.ViaMartingale.RevFiltration - - + + Exchangeability.DeFinetti.ViaMartingale.LocalInfrastructure - -Exchangeability.DeFinetti.ViaMartingale.LocalInfrastructure + +Exchangeability.DeFinetti.ViaMartingale.LocalInfrastructure Exchangeability.DeFinetti.ViaMartingale.LocalInfrastructure->Exchangeability.DeFinetti.ViaMartingale.PairLawEquality - - + + Exchangeability.DeFinetti.TheoremViaMartingale - -Exchangeability.DeFinetti.TheoremViaMartingale + +Exchangeability.DeFinetti.TheoremViaMartingale Exchangeability.DeFinetti.ViaMartingale->Exchangeability.DeFinetti.TheoremViaMartingale - - + + Exchangeability.DeFinetti.CommonEnding->Exchangeability.DeFinetti.ViaMartingale - - + + Exchangeability.DeFinetti.CommonEnding->Exchangeability.DeFinetti.ViaKoopman.InfraCore - - + + - + Exchangeability.DeFinetti.TheoremViaL2 - -Exchangeability.DeFinetti.TheoremViaL2 + +Exchangeability.DeFinetti.TheoremViaL2 Exchangeability.DeFinetti.CommonEnding->Exchangeability.DeFinetti.TheoremViaL2 - - + + Exchangeability.DeFinetti.ViaMartingale.DirectingMeasure->Exchangeability.DeFinetti.ViaMartingale - - + + Exchangeability.DeFinetti.Theorem - -Exchangeability.DeFinetti.Theorem + +Exchangeability.DeFinetti.Theorem Exchangeability.DeFinetti.TheoremViaMartingale->Exchangeability.DeFinetti.Theorem - - + + - + Exchangeability.Ergodic.InvariantSigma - -Exchangeability.Ergodic.InvariantSigma + +Exchangeability.Ergodic.InvariantSigma Exchangeability.Ergodic.KoopmanMeanErgodic->Exchangeability.Ergodic.InvariantSigma - - + + - + Exchangeability - -Exchangeability + +Exchangeability Exchangeability.DeFinetti.Theorem->Exchangeability - - + + Exchangeability.DeFinetti.MartingaleHelpers - -Exchangeability.DeFinetti.MartingaleHelpers + +Exchangeability.DeFinetti.MartingaleHelpers Exchangeability.PathSpace.CylinderHelpers->Exchangeability.DeFinetti.MartingaleHelpers - - + + Exchangeability.Probability.InfiniteProduct - -Exchangeability.Probability.InfiniteProduct + +Exchangeability.Probability.InfiniteProduct Exchangeability.Probability.InfiniteProduct->Exchangeability.Contractability - - + + Exchangeability.DeFinetti.ViaKoopman.KoopmanCommutation - -Exchangeability.DeFinetti.ViaKoopman.KoopmanCommutation + +Exchangeability.DeFinetti.ViaKoopman.KoopmanCommutation - + Exchangeability.DeFinetti.ViaKoopman.CesaroL2ToL1 - -Exchangeability.DeFinetti.ViaKoopman.CesaroL2ToL1 + +Exchangeability.DeFinetti.ViaKoopman.CesaroL2ToL1 Exchangeability.DeFinetti.ViaKoopman.KoopmanCommutation->Exchangeability.DeFinetti.ViaKoopman.CesaroL2ToL1 - - + + Exchangeability.DeFinetti.ViaKoopman.CesaroHelpers - -Exchangeability.DeFinetti.ViaKoopman.CesaroHelpers + +Exchangeability.DeFinetti.ViaKoopman.CesaroHelpers Exchangeability.DeFinetti.ViaKoopman.CesaroHelpers->Exchangeability.DeFinetti.ViaKoopman.KoopmanCommutation - - + + Exchangeability.DeFinetti.ViaKoopman.CylinderFunctions->Exchangeability.DeFinetti.ViaKoopman.KoopmanCommutation - - + + Exchangeability.DeFinetti.ViaKoopman.InfraLagConstancy - -Exchangeability.DeFinetti.ViaKoopman.InfraLagConstancy + +Exchangeability.DeFinetti.ViaKoopman.InfraLagConstancy - + Exchangeability.DeFinetti.ViaKoopman.InfraGeneralized - -Exchangeability.DeFinetti.ViaKoopman.InfraGeneralized + +Exchangeability.DeFinetti.ViaKoopman.InfraGeneralized Exchangeability.DeFinetti.ViaKoopman.InfraLagConstancy->Exchangeability.DeFinetti.ViaKoopman.InfraGeneralized - - + + Exchangeability.DeFinetti.ViaKoopman.InfraCore->Exchangeability.DeFinetti.ViaKoopman.InfraLagConstancy - - - - - - -Exchangeability.Probability.CenteredVariables - -Exchangeability.Probability.CenteredVariables + + - Exchangeability.Probability.CenteredVariables->Exchangeability.DeFinetti.ViaL2.CesaroConvergence.Cauchy - - + + - + Exchangeability.DeFinetti.ViaL2.CesaroConvergence.KallenbergBound - -Exchangeability.DeFinetti.ViaL2.CesaroConvergence.KallenbergBound + +Exchangeability.DeFinetti.ViaL2.CesaroConvergence.KallenbergBound Exchangeability.Probability.CenteredVariables->Exchangeability.DeFinetti.ViaL2.CesaroConvergence.KallenbergBound - - + + Exchangeability.Util.FinsetHelpers - -Exchangeability.Util.FinsetHelpers + +Exchangeability.Util.FinsetHelpers Exchangeability.Util.FinsetHelpers->Exchangeability.DeFinetti.ViaL2.CesaroConvergence.Cauchy - - + + Exchangeability.DeFinetti.ViaL2.DirectingMeasureBridge - -Exchangeability.DeFinetti.ViaL2.DirectingMeasureBridge + +Exchangeability.DeFinetti.ViaL2.DirectingMeasureBridge Exchangeability.DeFinetti.ViaL2.DirectingMeasureIntegral - -Exchangeability.DeFinetti.ViaL2.DirectingMeasureIntegral + +Exchangeability.DeFinetti.ViaL2.DirectingMeasureIntegral Exchangeability.DeFinetti.ViaL2.DirectingMeasureBridge->Exchangeability.DeFinetti.ViaL2.DirectingMeasureIntegral - - + + Exchangeability.DeFinetti.ViaL2.DirectingMeasureIic - -Exchangeability.DeFinetti.ViaL2.DirectingMeasureIic + +Exchangeability.DeFinetti.ViaL2.DirectingMeasureIic Exchangeability.DeFinetti.ViaL2.DirectingMeasureIic->Exchangeability.DeFinetti.ViaL2.DirectingMeasureBridge - - + + Exchangeability.Probability.LpNormHelpers - -Exchangeability.Probability.LpNormHelpers + +Exchangeability.Probability.LpNormHelpers + + +Exchangeability.Probability.LpNormHelpers->Exchangeability.DeFinetti.ViaKoopman.CesaroHelpers + + + Exchangeability.Probability.LpNormHelpers->Exchangeability.Probability.CenteredVariables - - + + Exchangeability.DeFinetti.ViaMartingale.IndicatorAlgebra - -Exchangeability.DeFinetti.ViaMartingale.IndicatorAlgebra + +Exchangeability.DeFinetti.ViaMartingale.IndicatorAlgebra Exchangeability.DeFinetti.ViaMartingale.IndicatorAlgebra->Exchangeability.DeFinetti.ViaMartingale.Factorization - - + + Exchangeability.DeFinetti.MartingaleHelpers->Exchangeability.DeFinetti.ViaMartingale.ShiftOperations - - + + Exchangeability.DeFinetti.MartingaleHelpers->Exchangeability.DeFinetti.ViaMartingale.LocalInfrastructure - - + + Exchangeability.DeFinetti.MartingaleHelpers->Exchangeability.DeFinetti.ViaKoopman.InfraCore - - + + Exchangeability.DeFinetti.MartingaleHelpers->Exchangeability.DeFinetti.ViaMartingale.IndicatorAlgebra - - + + Exchangeability.DeFinetti.ViaL2.AlphaConvergence.AlphaIicEq - -Exchangeability.DeFinetti.ViaL2.AlphaConvergence.AlphaIicEq + +Exchangeability.DeFinetti.ViaL2.AlphaConvergence.AlphaIicEq - + Exchangeability.DeFinetti.ViaL2.AlphaConvergence - -Exchangeability.DeFinetti.ViaL2.AlphaConvergence + +Exchangeability.DeFinetti.ViaL2.AlphaConvergence Exchangeability.DeFinetti.ViaL2.AlphaConvergence.AlphaIicEq->Exchangeability.DeFinetti.ViaL2.AlphaConvergence - - + + Exchangeability.DeFinetti.ViaL2.AlphaIicCE - -Exchangeability.DeFinetti.ViaL2.AlphaIicCE + +Exchangeability.DeFinetti.ViaL2.AlphaIicCE Exchangeability.DeFinetti.ViaL2.AlphaIicCE->Exchangeability.DeFinetti.ViaL2.AlphaConvergence.AlphaIicEq - - + + Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointCommon - -Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointCommon + +Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointCommon Exchangeability.DeFinetti.ViaL2.AlphaIicCE->Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointCommon - - + + Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L1->Exchangeability.DeFinetti.ViaL2.AlphaConvergence.AlphaIicEq - - + + Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtBot - -Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtBot + +Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtBot Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L1->Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtBot - - + + Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtTop - -Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtTop + +Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtTop Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L1->Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtTop - - + + - + Exchangeability.DeFinetti.ViaL2.CesaroConvergence - -Exchangeability.DeFinetti.ViaL2.CesaroConvergence + +Exchangeability.DeFinetti.ViaL2.CesaroConvergence Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L1->Exchangeability.DeFinetti.ViaL2.CesaroConvergence - - + + - + Exchangeability.DeFinetti.ViaL2 - -Exchangeability.DeFinetti.ViaL2 + +Exchangeability.DeFinetti.ViaL2 Exchangeability.DeFinetti.ViaL2.MoreL2Helpers->Exchangeability.DeFinetti.ViaL2 - - + + Exchangeability.DeFinetti.ViaL2.DirectingMeasureIntegral->Exchangeability.DeFinetti.ViaL2.MoreL2Helpers - - + + Exchangeability.Tail.CondExpShiftInvariance->Exchangeability.DeFinetti.ViaL2.MoreL2Helpers - - + + Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtBot->Exchangeability.DeFinetti.ViaL2.AlphaConvergence - - + + Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointCommon->Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtBot - - + + Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointCommon->Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtTop - - + + Exchangeability.DeFinetti.ViaL2.AlphaConvergence.EndpointAtTop->Exchangeability.DeFinetti.ViaL2.AlphaConvergence - - + + Exchangeability.DeFinetti.ViaL2.BlockAverages.TwoWindows - -Exchangeability.DeFinetti.ViaL2.BlockAverages.TwoWindows + +Exchangeability.DeFinetti.ViaL2.BlockAverages.TwoWindows Exchangeability.DeFinetti.ViaL2.BlockAverages.TwoWindows->Exchangeability.DeFinetti.ViaL2.BlockAverages - - + + Exchangeability.DeFinetti.ViaL2.BlockAverages.LongVsTail - -Exchangeability.DeFinetti.ViaL2.BlockAverages.LongVsTail + +Exchangeability.DeFinetti.ViaL2.BlockAverages.LongVsTail Exchangeability.DeFinetti.ViaL2.BlockAverages.LongVsTail->Exchangeability.DeFinetti.ViaL2.BlockAverages - - + + Exchangeability.DeFinetti.L2Helpers - -Exchangeability.DeFinetti.L2Helpers + +Exchangeability.DeFinetti.L2Helpers Exchangeability.DeFinetti.L2Helpers->Exchangeability.Probability.CenteredVariables - - + + - + Exchangeability.DeFinetti.ViaL2.BlockAverages.Covariance - -Exchangeability.DeFinetti.ViaL2.BlockAverages.Covariance + +Exchangeability.DeFinetti.ViaL2.BlockAverages.Covariance Exchangeability.DeFinetti.L2Helpers->Exchangeability.DeFinetti.ViaL2.BlockAverages.Covariance - - - - - - -Exchangeability.Probability.IntegrationHelpers - -Exchangeability.Probability.IntegrationHelpers - - - - -Exchangeability.Probability.IntegrationHelpers->Exchangeability.Probability.CenteredVariables - - - - - - -Exchangeability.Probability.CondExp - -Exchangeability.Probability.CondExp - - - - -Exchangeability.Probability.IntegrationHelpers->Exchangeability.Probability.CondExp - - + + - + Exchangeability.DeFinetti.ViaKoopman.Infrastructure - -Exchangeability.DeFinetti.ViaKoopman.Infrastructure + +Exchangeability.DeFinetti.ViaKoopman.Infrastructure Exchangeability.DeFinetti.ViaKoopman.Infrastructure->Exchangeability.DeFinetti.ViaKoopman.CesaroHelpers - - - - - - -Exchangeability.DeFinetti.ViaKoopman.LpCondExpHelpers - -Exchangeability.DeFinetti.ViaKoopman.LpCondExpHelpers - - - - -Exchangeability.DeFinetti.ViaKoopman.LpCondExpHelpers->Exchangeability.DeFinetti.ViaKoopman.CesaroHelpers - - + + Exchangeability.DeFinetti.ViaL2.CesaroConvergence->Exchangeability.DeFinetti.ViaL2 - - + + Exchangeability.DeFinetti.ViaL2.CesaroConvergence.KallenbergBound->Exchangeability.DeFinetti.ViaL2.CesaroConvergence - - + + Exchangeability.DeFinetti.ViaL2.BlockAverages.Covariance->Exchangeability.DeFinetti.ViaL2.BlockAverages.TwoWindows - - + + Exchangeability.DeFinetti.ViaL2.BlockAverages.Covariance->Exchangeability.DeFinetti.ViaL2.BlockAverages.LongVsTail - - + + Exchangeability.Contractability->Exchangeability.Core - - + + Exchangeability.Contractability->Exchangeability.DeFinetti.ViaMartingale.ShiftOperations - - + + Exchangeability.Contractability->Exchangeability.DeFinetti.L2Helpers - - + + - + Exchangeability.ConditionallyIID - -Exchangeability.ConditionallyIID + +Exchangeability.ConditionallyIID Exchangeability.Contractability->Exchangeability.ConditionallyIID - - + + Exchangeability.Probability.CondExp->Exchangeability.Probability.CondIndep.Basic - - + + Exchangeability.Probability.CondExp->Exchangeability.DeFinetti.ViaMartingale.FutureFiltration - - + + Exchangeability.Probability.CondExp->Exchangeability.DeFinetti.ViaKoopman.InfraCore - - + + Exchangeability.Probability.CondExp->Exchangeability.DeFinetti.ViaL2.MainConvergence - - + + Exchangeability.DeFinetti.ViaKoopman.InfraGeneralized->Exchangeability.DeFinetti.ViaKoopman.Infrastructure - - + + - + Exchangeability.DeFinetti.ViaKoopman.CesaroL1Bounded - -Exchangeability.DeFinetti.ViaKoopman.CesaroL1Bounded + +Exchangeability.DeFinetti.ViaKoopman.CesaroL1Bounded Exchangeability.DeFinetti.ViaKoopman.CesaroL2ToL1->Exchangeability.DeFinetti.ViaKoopman.CesaroL1Bounded - - + + Exchangeability.ConditionallyIID->Exchangeability.DeFinetti.CommonEnding - - + + Exchangeability.ConditionallyIID->Exchangeability.DeFinetti.ViaL2.MainConvergence - - + + - + Exchangeability.Probability.MeasureKernels - -Exchangeability.Probability.MeasureKernels + +Exchangeability.Probability.MeasureKernels Exchangeability.Probability.MeasureKernels->Exchangeability.DeFinetti.ViaMartingale.FiniteProduct - - + + Exchangeability.Probability.MeasureKernels->Exchangeability.ConditionallyIID - - + + Exchangeability.DeFinetti.ViaMartingale.RevFiltration->Exchangeability.DeFinetti.ViaMartingale.FutureFiltration - - + + Exchangeability.DeFinetti.ViaMartingale.RevFiltration->Exchangeability.DeFinetti.ViaMartingale.KallenbergChain - - + + Exchangeability.DeFinetti.TheoremViaL2->Exchangeability - - + + Exchangeability.DeFinetti.ViaL2->Exchangeability.DeFinetti.TheoremViaL2 - - + + - + Exchangeability.DeFinetti.ViaKoopman.CesaroPairFactorization - -Exchangeability.DeFinetti.ViaKoopman.CesaroPairFactorization + +Exchangeability.DeFinetti.ViaKoopman.CesaroPairFactorization Exchangeability.DeFinetti.ViaKoopman.CesaroL1Bounded->Exchangeability.DeFinetti.ViaKoopman.CesaroPairFactorization - - + + Exchangeability.Probability.Martingale.Crossings->Exchangeability.Probability.Martingale.Convergence - - + + - + Exchangeability.DeFinetti.ViaL2.WindowMachinery - -Exchangeability.DeFinetti.ViaL2.WindowMachinery + +Exchangeability.DeFinetti.ViaL2.WindowMachinery Exchangeability.DeFinetti.ViaL2.WindowMachinery->Exchangeability.DeFinetti.ViaL2.BlockAverages.TwoWindows - - + + - + Exchangeability.DeFinetti.ViaL2.DirectingMeasureCore - -Exchangeability.DeFinetti.ViaL2.DirectingMeasureCore + +Exchangeability.DeFinetti.ViaL2.DirectingMeasureCore Exchangeability.DeFinetti.ViaL2.DirectingMeasureCore->Exchangeability.DeFinetti.ViaL2.DirectingMeasureIic - - + + Exchangeability.DeFinetti.ViaL2.AlphaConvergence->Exchangeability.DeFinetti.ViaL2.DirectingMeasureCore - - + + - + Exchangeability.Probability.Martingale.Reverse - -Exchangeability.Probability.Martingale.Reverse + +Exchangeability.Probability.Martingale.Reverse Exchangeability.Probability.Martingale.Reverse->Exchangeability.Probability.Martingale.Crossings.Bounds - - + + Exchangeability.Bridge.CesaroToCondExp->Exchangeability.DeFinetti.TheoremViaKoopman - - + + Exchangeability.Probability.TripleLawDropInfo.DropInfo->Exchangeability.DeFinetti.ViaMartingale.KallenbergChain - - + + Exchangeability.Probability.TripleLawDropInfo.DropInfo->Exchangeability.Probability.TripleLawDropInfo - - + + Exchangeability.Probability.CondIndep->Exchangeability.DeFinetti.ViaMartingale.LocalInfrastructure - - + + Exchangeability.Ergodic.ShiftInvariantRepresentatives->Exchangeability.Ergodic.InvariantSigma - - + + Exchangeability.DeFinetti.ViaMartingale.Factorization->Exchangeability.DeFinetti.ViaMartingale.FiniteProduct - - + + Exchangeability.Ergodic.InvariantSigma->Exchangeability.DeFinetti.ViaKoopman.InfraCore - - + + Exchangeability.DeFinetti.TheoremViaKoopman->Exchangeability - - + + Exchangeability.DeFinetti.ViaKoopman.CesaroPairFactorization->Exchangeability.DeFinetti.ViaKoopman.CesaroConvergence - - + + - + Exchangeability.Ergodic.ProjectionLemmas - -Exchangeability.Ergodic.ProjectionLemmas + +Exchangeability.Ergodic.ProjectionLemmas Exchangeability.Ergodic.ProjectionLemmas->Exchangeability.Ergodic.InvariantSigma - - + + - + Exchangeability.DeFinetti.ViaL2.AlphaIic - -Exchangeability.DeFinetti.ViaL2.AlphaIic + +Exchangeability.DeFinetti.ViaL2.AlphaIic Exchangeability.DeFinetti.ViaL2.MainConvergence->Exchangeability.DeFinetti.ViaL2.AlphaIic - - + + Exchangeability.DeFinetti.ViaL2.AlphaIic->Exchangeability.DeFinetti.ViaL2.AlphaIicCE - - + + diff --git a/home_page/index.md b/home_page/index.md index be8776c5..cdd67dec 100644 --- a/home_page/index.md +++ b/home_page/index.md @@ -35,7 +35,7 @@ We formalize **all three proofs** from Kallenberg (2005): | Graph | Description | |-------|-------------| -| [Modules (SVG)]({{ site.url }}/blueprint/import_graph_colored.svg) | 105 files, clickable links to docs | +| [Modules (SVG)]({{ site.url }}/blueprint/import_graph_colored.svg) | 103 files, clickable links to docs | | [Modules (interactive)]({{ site.url }}/blueprint/import_graph_colored.html) | Draggable, zoomable | | [All declarations]({{ site.url }}/blueprint/import_graph_full_declarations.html) | Searchable | | [Blueprint only]({{ site.url }}/blueprint/dep_graph_document.html) | Documented items | From 5132b45eb47dff2a842e611f5f2778198a70ebbc Mon Sep 17 00:00:00 2001 From: Cameron Freer Date: Tue, 9 Jun 2026 23:45:28 +0000 Subject: [PATCH 3/6] style: golf 4 proof sites (term-mode, inline single-use haves) - Contractability.lean: strictMono_all_lt_succ_last to term mode; inline h_mono into the calc; inline hf/hg into the obtain - Core.lean: cylinder_subset_prefixCylinders uses the definitional equality directly instead of an explicit rfl-transport Statements unchanged; remaining find_golfable candidates skipped where the existing structure is clearer. --- Exchangeability/Contractability.lean | 14 +++++--------- Exchangeability/Core.lean | 12 ++++-------- 2 files changed, 9 insertions(+), 17 deletions(-) diff --git a/Exchangeability/Contractability.lean b/Exchangeability/Contractability.lean index fa1dba7c..304ea3d3 100644 --- a/Exchangeability/Contractability.lean +++ b/Exchangeability/Contractability.lean @@ -290,10 +290,8 @@ lemma exists_perm_extending_strictMono {m n : ℕ} (k : Fin m → ℕ) -- and `g` = the strictly monotone embedding via `k`. let f : Fin m → Fin n := Fin.castLE hmn let g : Fin m → Fin n := fun i => ⟨k i, hk_bound i⟩ - have hf : Function.Injective f := Fin.castLE_injective hmn - have hg : Function.Injective g := fun i j hij => - hk_mono.injective (Fin.mk.inj hij) - obtain ⟨σ, hσ⟩ := Equiv.Perm.exists_extending_pair f g hf hg + obtain ⟨σ, hσ⟩ := Equiv.Perm.exists_extending_pair f g (Fin.castLE_injective hmn) + (fun i j hij => hk_mono.injective (Fin.mk.inj hij)) exact ⟨σ, fun i => congrArg Fin.val (hσ i)⟩ /- Helper: relabeling coordinates by a finite permutation is measurable as a map @@ -385,9 +383,8 @@ This follows from monotonicity and the fact that any `i` is at most `last`. -/ private lemma strictMono_all_lt_succ_last {m : ℕ} (k : Fin m → ℕ) (hk : StrictMono k) (i : Fin m) (last : Fin m) (h_last : ∀ j, j ≤ last) : - k i ≤ k last := by - apply StrictMono.monotone hk - exact h_last i + k i ≤ k last := + hk.monotone (h_last i) /-- Helper lemma: The length of the domain is bounded by the maximum value plus one. @@ -398,9 +395,8 @@ This uses the fact that strictly monotone functions satisfy `i ≤ k(i)` for all private lemma strictMono_length_le_max_succ {m : ℕ} (k : Fin m → ℕ) (hk : StrictMono k) (last : Fin m) (h_last_is_max : last.val + 1 = m) : m ≤ k last + 1 := by - have h_mono : last.val ≤ k last := strictMono_Fin_ge_id hk last calc m = last.val + 1 := h_last_is_max.symm - _ ≤ k last + 1 := Nat.add_le_add_right h_mono 1 + _ ≤ k last + 1 := Nat.add_le_add_right (strictMono_Fin_ge_id hk last) 1 /-- **Main theorem:** Every exchangeable sequence is contractable. diff --git a/Exchangeability/Core.lean b/Exchangeability/Core.lean index 4912dbc5..82e924ad 100644 --- a/Exchangeability/Core.lean +++ b/Exchangeability/Core.lean @@ -208,15 +208,11 @@ private lemma cylinder_subset_prefixCylinders {s : Finset ℕ} {S : Set (∀ _ : intro i hi have hle : i ≤ s.sup id := by convert Finset.le_sup (f := id) hi omega - -- Transport `S` along the inclusion into the initial segment. - let ι : s → Fin N := fun x => ⟨x.1, h_mem x.1 x.2⟩ - let pull : (Fin N → α) → (∀ i : s, α) := fun x => fun y => x (ι y) + -- Transport `S` along the inclusion into the initial segment; the cylinder is + -- definitionally the prefix cylinder of the pulled-back set. + let pull : (Fin N → α) → (∀ i : s, α) := fun x => fun y => x ⟨y.1, h_mem y.1 y.2⟩ have hpull_meas : Measurable pull := by fun_prop - have hs_eq : - MeasureTheory.cylinder (α:=fun _ : ℕ => α) s S = - prefixCylinder (α:=α) (pull ⁻¹' S) := rfl - refine hs_eq ▸ prefixCylinder_mem_prefixCylinders (α:=α) ?_ - exact hpull_meas hS + exact prefixCylinder_mem_prefixCylinders (α:=α) (hpull_meas hS) /-- The σ-algebra generated by prefix cylinders is the product σ-algebra. From 0f3ee873e3936d8f662b6516e095b02a9f113618 Mon Sep 17 00:00:00 2001 From: Cameron Freer Date: Tue, 9 Jun 2026 23:47:47 +0000 Subject: [PATCH 4/6] docs+ci: add Phase 11 to chronology; add exact-match axiom check to CI - DEVELOPMENT_CHRONOLOGY.md: header recounted from git (4,247 commits, 3,373 non-merge, through June 8 2026); new Phase 11 covering the Jan 9 - June 8 maintenance period - blueprint.yml: after the build, #print axioms output for deFinetti is diffed against the expected single line, so any added axiom fails CI --- .github/workflows/blueprint.yml | 9 +++++++++ DEVELOPMENT_CHRONOLOGY.md | 22 +++++++++++++++++++++- 2 files changed, 30 insertions(+), 1 deletion(-) diff --git a/.github/workflows/blueprint.yml b/.github/workflows/blueprint.yml index 22bdc710..4a721bd5 100644 --- a/.github/workflows/blueprint.yml +++ b/.github/workflows/blueprint.yml @@ -44,6 +44,15 @@ jobs: run: | lake build + - name: Check axioms + run: | + printf 'import Exchangeability\n#print axioms Exchangeability.DeFinetti.deFinetti\n' \ + | lake env lean --stdin > /tmp/axioms.out + # Exact full-output comparison: an extra axiom or extra line fails the step. + diff /tmp/axioms.out - <<'EOF' + 'Exchangeability.DeFinetti.deFinetti' depends on axioms: [propext, Classical.choice, Quot.sound] + EOF + - name: Check declarations run: | ~/.elan/bin/lake exe checkdecls blueprint/lean_decls diff --git a/DEVELOPMENT_CHRONOLOGY.md b/DEVELOPMENT_CHRONOLOGY.md index 57a5f6bb..26f9804f 100644 --- a/DEVELOPMENT_CHRONOLOGY.md +++ b/DEVELOPMENT_CHRONOLOGY.md @@ -1,6 +1,7 @@ # Development Chronology -A narrative history of this Lean 4 formalization, based on 4,024 commits (3,166 non-merge) from Sept 29, 2025 - Jan 8, 2026. +A narrative history of this Lean 4 formalization, based on 4,247 commits (3,373 non-merge, +counted on `main` including merges in the total) from Sept 29, 2025 - June 8, 2026. ## Phase 0: Project Initialization (Sept 29 - Oct 1, 2025) @@ -150,6 +151,25 @@ Created `DirectingMeasure.lean` infrastructure for the identification chain: a s 2. Prepares for potential mathlib integration 3. Professional code quality standards +## Phase 11: Maintenance & Hardening (Jan 9 - June 8, 2026) + +**172 commits (165 non-merge) of post-completion upkeep** + +- `886550af` Toolchain upgrade to Lean 4.27.0-rc1, then `8ab88df2` to v4.30.0-rc2 with + matching mathlib +- Cross-tree dead-code sweeps, e.g. `b642382e` (−1,713 lines) and follow-ups — several + thousand lines of abandoned scaffolding removed +- Linter hygiene: `3c40bcbf` cleared all 44 remaining linter warnings; suppressions pruned + in stages until `3e7b30d7` dropped the last ones (0 remaining) +- Term-mode golf batches, e.g. `ef4e47b1` collapsing 32 `by exact` wrappers +- `93961e6f` Canonical `Contractable.{comp, map_pair}` API; call sites migrated and the old + L2Helpers wrappers dropped +- Blueprint/CI: `8f848259` PR-triggered blueprint consistency check; web + pdf artifact + refreshes; `289affe8` doc count/date refresh ahead of the v1.3 release + +**Pattern:** With all three proofs complete, work shifted from proving to curation — +keeping the tree warning-free, current with mathlib, and honest in its documentation. + --- ## Key Patterns From 0a308c78a77db76b656e70bed4733b8843d66ec0 Mon Sep 17 00:00:00 2001 From: Cameron Freer Date: Tue, 9 Jun 2026 23:51:09 +0000 Subject: [PATCH 5/6] docs(STATUS): refresh LOC table after helper consolidation --- STATUS.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/STATUS.md b/STATUS.md index 9d8b33e6..f6f93b46 100644 --- a/STATUS.md +++ b/STATUS.md @@ -1,6 +1,6 @@ # Project Status: de Finetti Theorem Formalization -**Last Updated:** 2026-05-27 +**Last Updated:** 2026-06-09 ## Executive Summary @@ -10,10 +10,10 @@ LOC counts below cover the `Exchangeability/` library tree (excluding the top-le | Proof Approach | Lines | Build | Status | |---------------|------:|-------|--------| | **ViaMartingale** | 2,804 | Pass | **COMPLETE** | -| **ViaL2** | 7,444 | Pass | **COMPLETE** | -| **ViaKoopman** | 5,402 | Pass | **COMPLETE** | -| *Shared infrastructure* | 11,117 | — | — | -| **Total** | **26,767** | Pass | **COMPLETE** | +| **ViaL2** | 7,443 | Pass | **COMPLETE** | +| **ViaKoopman** | 5,305 | Pass | **COMPLETE** | +| *Shared infrastructure* | 11,111 | — | — | +| **Total** | **26,663** | Pass | **COMPLETE** | **All three proofs complete.** From 88783b2e81bc2c1bbac5837f2a52f825c4a64a6f Mon Sep 17 00:00:00 2001 From: Cameron Freer Date: Wed, 10 Jun 2026 00:19:21 +0000 Subject: [PATCH 6/6] fix: refresh stale graph HTML after module deletions; pin Phase 11 count convention - import_graph_colored.html: re-embed the GEXF from the current import graph (lake exe graph; 103 modules, includes the new LpNormHelpers->CesaroHelpers and IntegrationHelpers->AntitoneLimit edges). The page's own JS computes colors, so the proof-palette view is unchanged. - import_graph_full_declarations.html: drop the 5 declaration nodes (and their 5 edges) for the deleted ViaKoopman/LpCondExpHelpers from the Feb snapshot; arrays re-validated (426 nodes / 468 edges, no dangling ids). The live copy is rebuilt by docgen CI on push to main. - DEVELOPMENT_CHRONOLOGY.md: Phase 11 count was time-of-day dependent (bare --since date); now 226 commits (208 non-merge) with the explicit --since="2026-01-09 00:00" convention stated inline. --- DEVELOPMENT_CHRONOLOGY.md | 2 +- blueprint/web/import_graph_colored.html | 2 +- blueprint/web/import_graph_full_declarations.html | 4 ++-- 3 files changed, 4 insertions(+), 4 deletions(-) diff --git a/DEVELOPMENT_CHRONOLOGY.md b/DEVELOPMENT_CHRONOLOGY.md index 26f9804f..68f70ada 100644 --- a/DEVELOPMENT_CHRONOLOGY.md +++ b/DEVELOPMENT_CHRONOLOGY.md @@ -153,7 +153,7 @@ Created `DirectingMeasure.lean` infrastructure for the identification chain: a s ## Phase 11: Maintenance & Hardening (Jan 9 - June 8, 2026) -**172 commits (165 non-merge) of post-completion upkeep** +**226 commits (208 non-merge, `--since="2026-01-09 00:00"` on `main`) of post-completion upkeep** - `886550af` Toolchain upgrade to Lean 4.27.0-rc1, then `8ab88df2` to v4.30.0-rc2 with matching mathlib diff --git a/blueprint/web/import_graph_colored.html b/blueprint/web/import_graph_colored.html index 5c743d76..f5ce0a05 100644 --- a/blueprint/web/import_graph_colored.html +++ b/blueprint/web/import_graph_colored.html @@ -501,7 +501,7 @@

Import Graph for Exchangeability

// NOTE: currently the Lean script searches for this line to replace it (exact search)! // If you modify it, modify the Lean script (ImportGraph/Cli.lean) accordingly! -render_gexf(" Lean ImportGraph ") +render_gexf(" Lean ImportGraph ") container.addEventListener("dragover", ev => { event.preventDefault(); diff --git a/blueprint/web/import_graph_full_declarations.html b/blueprint/web/import_graph_full_declarations.html index 14ad6b7f..6feb594c 100644 --- a/blueprint/web/import_graph_full_declarations.html +++ b/blueprint/web/import_graph_full_declarations.html @@ -32,8 +32,8 @@
Large nodes = key theorems