From 7e52cc4e6286efc476b932901ec70838a41ae715 Mon Sep 17 00:00:00 2001 From: Cameron Freer Date: Wed, 10 Jun 2026 20:44:52 +0000 Subject: [PATCH 1/5] refactor: replace 4 helpers with mathlib v4.30.0 equivalents MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - Finset.filter_val_lt_card -> Fin.card_filter_val_lt (2 sites); Util/FinsetHelpers.lean deleted (sole content) - integrable_of_bounded (Probability/CondExp.lean) -> Integrable.of_bound (6 sites in CesaroL1Bounded) - integrable_of_bounded_measurable (LpNormHelpers) -> Integrable.of_bound (10 sites across ViaKoopman) - memLp_two_of_bounded (LpNormHelpers) -> MemLp.of_bound (3 sites) NOT migrated (audit claim didn't survive contact with the real call sites): aestronglyMeasurable_sub_of_tendsto_ae stays — mathlib's aestronglyMeasurable_of_tendsto_ae only covers the m = m0 case, while this helper's purpose is AEStronglyMeasurable[m] for m < m0 via a limsup witness; its docstring now records that. --- .../DeFinetti/ViaKoopman/BlockAverage.lean | 3 +- .../DeFinetti/ViaKoopman/CesaroHelpers.lean | 30 +++++++----- .../DeFinetti/ViaKoopman/CesaroL1Bounded.lean | 33 +++++++------ .../ViaKoopman/CesaroPairFactorization.lean | 11 +++-- .../ViaL2/CesaroConvergence/Cauchy.lean | 10 ++-- .../Probability/CenteredVariables.lean | 10 ++-- Exchangeability/Probability/CondExp.lean | 10 ---- .../Probability/LpNormHelpers.lean | 24 --------- .../Probability/SigmaAlgebraHelpers.lean | 7 ++- Exchangeability/Util/FinsetHelpers.lean | 49 ------------------- 10 files changed, 58 insertions(+), 129 deletions(-) delete mode 100644 Exchangeability/Util/FinsetHelpers.lean diff --git a/Exchangeability/DeFinetti/ViaKoopman/BlockAverage.lean b/Exchangeability/DeFinetti/ViaKoopman/BlockAverage.lean index 5a33077e..373116f3 100644 --- a/Exchangeability/DeFinetti/ViaKoopman/BlockAverage.lean +++ b/Exchangeability/DeFinetti/ViaKoopman/BlockAverage.lean @@ -173,7 +173,8 @@ lemma blockAvg_tendsto_condExp -- Key fact 2: Y is shift-invariant (CE w.r.t. mSI is constant on shift orbits) have hf_int : Integrable (fun ω : Ω[α] => f (ω 0)) μ := let ⟨C, hC⟩ := hf_bd - integrable_of_bounded_measurable (hf.comp (measurable_pi_apply 0)) C fun ω => hC (ω 0) + Integrable.of_bound (hf.comp (measurable_pi_apply 0)).aestronglyMeasurable C + (ae_of_all _ fun ω => hC (ω 0)) have h_Y_shift_inv : ∀ p : ℕ, (fun ω => Y (shift^[p] ω)) =ᵐ[μ] Y := by intro p diff --git a/Exchangeability/DeFinetti/ViaKoopman/CesaroHelpers.lean b/Exchangeability/DeFinetti/ViaKoopman/CesaroHelpers.lean index e9dfcf4a..e9942020 100644 --- a/Exchangeability/DeFinetti/ViaKoopman/CesaroHelpers.lean +++ b/Exchangeability/DeFinetti/ViaKoopman/CesaroHelpers.lean @@ -193,16 +193,18 @@ lemma cesaro_ce_eq_condexp (fun ω => (Finset.range (n + 1)).sum (fun j => μ[(fun ω => g (ω j)) | mSI] ω)) := by have hint : ∀ j ∈ Finset.range (n + 1), Integrable (fun ω => g (ω j)) μ := fun j _ => - hg_bd.elim fun Cg hCg => integrable_of_bounded_measurable - (hg_meas.comp (measurable_pi_apply j)) Cg (fun ω => hCg (ω j)) + hg_bd.elim fun Cg hCg => Integrable.of_bound + (hg_meas.comp (measurable_pi_apply j)).aestronglyMeasurable Cg + (ae_of_all _ fun ω => hCg (ω j)) exact condExp_sum_finset (m := mSI) (_hm := hmSI) (Finset.range (n + 1)) (fun j ω => g (ω j)) hint -- Each term μ[g(ωⱼ)| mSI] =ᵐ μ[g(ω₀)| mSI] have h_term : ∀ j, μ[(fun ω => g (ω j)) | mSI] =ᵐ[μ] μ[(fun ω => g (ω 0)) | mSI] := fun j => by have hg_0_int : Integrable (fun ω => g (ω 0)) μ := - hg_bd.elim fun Cg hCg => integrable_of_bounded_measurable - (hg_meas.comp (measurable_pi_apply 0)) Cg (fun ω => hCg (ω 0)) + hg_bd.elim fun Cg hCg => Integrable.of_bound + (hg_meas.comp (measurable_pi_apply 0)).aestronglyMeasurable Cg + (ae_of_all _ fun ω => hCg (ω 0)) have h_shift : (fun ω => g (shift^[j] ω 0)) = (fun ω => g (ω j)) := by ext ω; simp only [shift_iterate_apply, zero_add] rw [← h_shift]; exact condexp_precomp_iterate_eq hσ hg_0_int @@ -355,10 +357,12 @@ lemma product_ce_constant_of_lag_const intro j _ obtain ⟨Cf, hCf⟩ := hf_bd obtain ⟨Cg, hCg⟩ := hg_bd - exact integrable_of_bounded_measurable - (hf_meas.comp (measurable_pi_apply 0) |>.mul (hg_meas.comp (measurable_pi_apply j))) - (Cf * Cg) - (fun ω => by simpa [abs_mul] using mul_le_mul (hCf (ω 0)) (hCg (ω j)) (abs_nonneg _) (le_trans (abs_nonneg _) (hCf (ω 0)))) + have hmeas : Measurable fun ω : Ω[α] => f (ω 0) * g (ω j) := + (hf_meas.comp (measurable_pi_apply 0)).mul (hg_meas.comp (measurable_pi_apply j)) + exact Integrable.of_bound hmeas.aestronglyMeasurable (Cf * Cg) + (ae_of_all _ fun ω => by + simpa [abs_mul, Real.norm_eq_abs] using mul_le_mul (hCf (ω 0)) (hCg (ω j)) + (abs_nonneg _) (le_trans (abs_nonneg _) (hCf (ω 0)))) exact condExp_sum_finset (shiftInvariantSigma_le (α := α)) (Finset.range (n + 1)) (fun j => fun ω => f (ω 0) * g (ω j)) hint @@ -494,10 +498,12 @@ lemma product_ce_constant_of_lag_const_from_one intro j _ obtain ⟨Cf, hCf⟩ := hf_bd obtain ⟨Cg, hCg⟩ := hg_bd - exact integrable_of_bounded_measurable - (hf_meas.comp (measurable_pi_apply 0) |>.mul (hg_meas.comp (measurable_pi_apply (j + 1)))) - (Cf * Cg) - (fun ω => by simpa [abs_mul] using mul_le_mul (hCf (ω 0)) (hCg (ω (j + 1))) (abs_nonneg _) (le_trans (abs_nonneg _) (hCf (ω 0)))) + have hmeas : Measurable fun ω : Ω[α] => f (ω 0) * g (ω (j + 1)) := + (hf_meas.comp (measurable_pi_apply 0)).mul (hg_meas.comp (measurable_pi_apply (j + 1))) + exact Integrable.of_bound hmeas.aestronglyMeasurable (Cf * Cg) + (ae_of_all _ fun ω => by + simpa [abs_mul, Real.norm_eq_abs] using mul_le_mul (hCf (ω 0)) (hCg (ω (j + 1))) + (abs_nonneg _) (le_trans (abs_nonneg _) (hCf (ω 0)))) exact condExp_sum_finset (shiftInvariantSigma_le (α := α)) (Finset.range n) (fun j => fun ω => f (ω 0) * g (ω (j + 1))) hint diff --git a/Exchangeability/DeFinetti/ViaKoopman/CesaroL1Bounded.lean b/Exchangeability/DeFinetti/ViaKoopman/CesaroL1Bounded.lean index ed24719d..19c165e8 100644 --- a/Exchangeability/DeFinetti/ViaKoopman/CesaroL1Bounded.lean +++ b/Exchangeability/DeFinetti/ViaKoopman/CesaroL1Bounded.lean @@ -198,16 +198,16 @@ lemma L1_cesaro_convergence have h_integrable_diff : Integrable (fun ω => g (ω 0) - g_M M (ω 0)) μ := by have h_g_M_int : Integrable (fun ω => g_M M (ω 0)) μ := by obtain ⟨C, hC⟩ := hg_M_bd M - refine Exchangeability.Probability.integrable_of_bounded ?_ ⟨C, fun ω => hC (ω 0)⟩ - exact (hg_M_meas M).comp (measurable_pi_apply 0) + exact Integrable.of_bound ((hg_M_meas M).comp (measurable_pi_apply 0)).aestronglyMeasurable + C (ae_of_all _ fun ω => hC (ω 0)) exact hg_int.sub h_g_M_int have h_ce_lin : μ[(fun ω => g (ω 0) - g_M M (ω 0)) | mSI] =ᵐ[μ] (fun ω => μ[(fun ω => g (ω 0)) | mSI] ω - μ[(fun ω => g_M M (ω 0)) | mSI] ω) := by have h_int_g : Integrable (fun ω => g (ω 0)) μ := hg_int have h_int_gM : Integrable (fun ω => g_M M (ω 0)) μ := by obtain ⟨C, hC⟩ := hg_M_bd M - refine Exchangeability.Probability.integrable_of_bounded ?_ ⟨C, fun ω => hC (ω 0)⟩ - exact (hg_M_meas M).comp (measurable_pi_apply 0) + exact Integrable.of_bound ((hg_M_meas M).comp (measurable_pi_apply 0)).aestronglyMeasurable + C (ae_of_all _ fun ω => hC (ω 0)) exact condExp_sub h_int_g h_int_gM mSI calc ∫ ω, |μ[(fun ω => g (ω 0)) | mSI] ω - μ[(fun ω => g_M M (ω 0)) | mSI] ω| ∂μ = ∫ ω, |μ[(fun ω => g (ω 0) - g_M M (ω 0)) | mSI] ω| ∂μ := by @@ -252,8 +252,8 @@ lemma L1_cesaro_convergence have h_int_ce_g : Integrable (μ[(fun ω => g (ω 0)) | mSI]) μ := integrable_condExp have h_int_gM : Integrable (fun ω => g_M M₀ (ω 0)) μ := by obtain ⟨C, hC⟩ := hg_M_bd M₀ - refine Exchangeability.Probability.integrable_of_bounded ?_ ⟨C, fun ω => hC (ω 0)⟩ - exact (hg_M_meas M₀).comp (measurable_pi_apply 0) + exact Integrable.of_bound ((hg_M_meas M₀).comp (measurable_pi_apply 0)).aestronglyMeasurable + C (ae_of_all _ fun ω => hC (ω 0)) have h_int_ce_gM : Integrable (μ[(fun ω => g_M M₀ (ω 0)) | mSI]) μ := integrable_condExp have h_int_A : Integrable (A n) μ := by @@ -275,8 +275,8 @@ lemma L1_cesaro_convergence have h_each_int : ∀ j ∈ Finset.range (n + 1), Integrable (fun ω => g_M M₀ (ω j)) μ := by intro j _ obtain ⟨C, hC⟩ := hg_M_bd M₀ - refine Exchangeability.Probability.integrable_of_bounded ?_ ⟨C, fun ω => hC (ω j)⟩ - exact (hg_M_meas M₀).comp (measurable_pi_apply j) + exact Integrable.of_bound ((hg_M_meas M₀).comp (measurable_pi_apply j)).aestronglyMeasurable + C (ae_of_all _ fun ω => hC (ω j)) exact integrable_finsetSum (Finset.range (n + 1)) h_each_int exact h_int_sum.const_mul (1 / ((n + 1) : ℝ)) @@ -322,8 +322,8 @@ lemma L1_cesaro_convergence exact (hσ.iterate j).integrable_comp_of_integrable hg_int have h_int_gMj : Integrable (fun ω => g_M M₀ (ω j)) μ := by obtain ⟨C, hC⟩ := hg_M_bd M₀ - refine Exchangeability.Probability.integrable_of_bounded ?_ ⟨C, fun ω => hC (ω j)⟩ - exact (hg_M_meas M₀).comp (measurable_pi_apply j) + exact Integrable.of_bound ((hg_M_meas M₀).comp (measurable_pi_apply j)).aestronglyMeasurable + C (ae_of_all _ fun ω => hC (ω j)) exact (h_int_gj.sub h_int_gMj).abs exact h_sum_int.const_mul (1 / ((n + 1) : ℝ)) · filter_upwards with ω @@ -340,8 +340,8 @@ lemma L1_cesaro_convergence exact (hσ.iterate j).integrable_comp_of_integrable hg_int have h_int_gMj : Integrable (fun ω => g_M M₀ (ω j)) μ := by obtain ⟨C, hC⟩ := hg_M_bd M₀ - refine Exchangeability.Probability.integrable_of_bounded ?_ ⟨C, fun ω => hC (ω j)⟩ - exact (hg_M_meas M₀).comp (measurable_pi_apply j) + exact Integrable.of_bound ((hg_M_meas M₀).comp (measurable_pi_apply j)).aestronglyMeasurable + C (ae_of_all _ fun ω => hC (ω j)) exact (h_int_gj.sub h_int_gMj).abs _ = (1 / (↑n + 1)) * ∑ j ∈ Finset.range (n + 1), ∫ ω, |g (ω 0) - g_M M₀ (ω 0)| ∂μ := by congr 1 @@ -407,12 +407,13 @@ lemma ce_lipschitz_convergence have hA_int : ∀ n, Integrable (A n) μ := fun n => (integrable_finsetSum (Finset.range (n + 1)) fun j _ => - integrable_of_bounded_measurable - (hg_meas.comp (measurable_pi_apply j)) Cg fun ω => hCg (ω j)).smul (1 / ((n + 1) : ℝ)) + Integrable.of_bound (f := fun ω : Ω[α] => g (ω j)) + (hg_meas.comp (measurable_pi_apply j)).aestronglyMeasurable Cg + (ae_of_all _ fun ω => hCg (ω j))).smul (1 / ((n + 1) : ℝ)) have hg0_int : Integrable (fun ω => g (ω 0)) μ := - integrable_of_bounded_measurable - (hg_meas.comp (measurable_pi_apply 0)) Cg (fun ω => hCg (ω 0)) + Integrable.of_bound (hg_meas.comp (measurable_pi_apply 0)).aestronglyMeasurable Cg + (ae_of_all _ fun ω => hCg (ω 0)) have hZ_int : ∀ n, Integrable (fun ω => f (ω 0) * A n ω) μ := fun n => integrable_mul_of_ae_bdd_left (hf_meas.comp (measurable_pi_apply 0)) diff --git a/Exchangeability/DeFinetti/ViaKoopman/CesaroPairFactorization.lean b/Exchangeability/DeFinetti/ViaKoopman/CesaroPairFactorization.lean index 7aab05cd..e07eaa23 100644 --- a/Exchangeability/DeFinetti/ViaKoopman/CesaroPairFactorization.lean +++ b/Exchangeability/DeFinetti/ViaKoopman/CesaroPairFactorization.lean @@ -108,8 +108,8 @@ private theorem h_tower_of_lagConst_from_one (1 / ((n + 1) : ℝ)) * (Finset.range (n + 1)).sum (fun j => g (ω j)) -- By L1_cesaro_convergence: A_n → Y in L¹ have hg_int : Integrable (fun ω => g (ω 0)) μ := - integrable_of_bounded_measurable - (hg_meas.comp (measurable_pi_apply 0)) Cg (fun ω => hCg (ω 0)) + Integrable.of_bound (hg_meas.comp (measurable_pi_apply 0)).aestronglyMeasurable Cg + (ae_of_all _ fun ω => hCg (ω 0)) have h_A_to_Y := L1_cesaro_convergence hσ g hg_meas hg_int -- A'_{n+1}(ω) = A_n(shift ω) have h_eq : ∀ n ω, A' (n + 1) ω = A n (shift ω) := by @@ -161,7 +161,8 @@ private theorem h_tower_of_lagConst_from_one - μ[(fun ω' => f (ω' 0) * Y ω') | mSI] ω| ∂μ) atTop (𝓝 0) := by -- Use ce_lipschitz_convergence with A' shifted by 1 have h_int : Integrable (fun ω => g (ω 0)) μ := - integrable_of_bounded_measurable (hg_meas.comp (measurable_pi_apply 0)) Cg (fun ω => hCg (ω 0)) + Integrable.of_bound (hg_meas.comp (measurable_pi_apply 0)).aestronglyMeasurable Cg + (ae_of_all _ fun ω => hCg (ω 0)) -- A'_{n+1} has the form (1/(n+1)) * Σ_{j=0}^n g(shift ω)_j = A_n(shift ω) -- Need to relate to ce_lipschitz_convergence format -- ce_lipschitz_convergence needs: A_n defined as (1/(n+1)) * Σ g(ω_j) @@ -178,8 +179,8 @@ private theorem h_tower_of_lagConst_from_one simp only [A', if_neg (Nat.ne_of_gt hm)] have h_sum : Integrable (fun ω => (Finset.range m).sum (fun j => g (ω (j + 1)))) μ := integrable_finsetSum (Finset.range m) (fun j _ => - integrable_of_bounded_measurable - (hg_meas.comp (measurable_pi_apply (j + 1))) Cg (fun ω => hCg (ω (j + 1)))) + Integrable.of_bound (hg_meas.comp (measurable_pi_apply (j + 1))).aestronglyMeasurable + Cg (ae_of_all _ fun ω => hCg (ω (j + 1)))) exact h_sum.smul (1 / (m : ℝ)) have hfA_int : Integrable (fun ω => f (ω 0) * A' (n + 1) ω) μ := integrable_mul_of_ae_bdd_left (hf_meas.comp (measurable_pi_apply 0)) diff --git a/Exchangeability/DeFinetti/ViaL2/CesaroConvergence/Cauchy.lean b/Exchangeability/DeFinetti/ViaL2/CesaroConvergence/Cauchy.lean index eb4077fa..f07e2eff 100644 --- a/Exchangeability/DeFinetti/ViaL2/CesaroConvergence/Cauchy.lean +++ b/Exchangeability/DeFinetti/ViaL2/CesaroConvergence/Cauchy.lean @@ -3,9 +3,9 @@ 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.Data.Fintype.Fin import Exchangeability.DeFinetti.ViaL2.BlockAvgDef import Exchangeability.Probability.CenteredVariables -import Exchangeability.Util.FinsetHelpers /-! # Cesàro Cauchy Property: `blockAvg_cauchy_in_L2` @@ -127,7 +127,7 @@ private lemma cesaro_cauchy_rho_lt rw [Finset.sum_const] _ = n • (n : ℝ)⁻¹ := by congr 1 - exact Finset.filter_val_lt_card (le_max_left n n') + simpa using Fin.card_filter_val_lt.trans (min_eq_right (le_max_left n n')) _ = 1 := by rw [nsmul_eq_mul] field_simp [Nat.cast_ne_zero.mpr (Nat.pos_iff_ne_zero.mp hn_pos)] @@ -156,7 +156,7 @@ private lemma cesaro_cauchy_rho_lt rw [Finset.sum_const] _ = n' • (n' : ℝ)⁻¹ := by congr 1 - exact Finset.filter_val_lt_card (le_max_right n n') + simpa using Fin.card_filter_val_lt.trans (min_eq_right (le_max_right n n')) _ = 1 := by rw [nsmul_eq_mul] field_simp [Nat.cast_ne_zero.mpr (Nat.pos_iff_ne_zero.mp hn'_pos)] @@ -184,8 +184,8 @@ private lemma cesaro_cauchy_rho_lt simp only [sub_zero, ξ] -- Z k.val is bounded, hence in L² -- Same proof as for Z 0: |Z k.val| ≤ |f| + |m| ≤ 1 + 1 = 2 - apply memLp_two_of_bounded (hZ_meas k.val) - intro ω + refine MemLp.of_bound (hZ_meas k.val).aestronglyMeasurable 2 (ae_of_all _ fun ω => ?_) + rw [Real.norm_eq_abs] -- Unfold ξ and Z to show |f(X k.val ω) - m| ≤ 2 have h1 : |f (X k.val ω)| ≤ 1 := hf_bdd (X k.val ω) have h2 : |∫ ω', f (X 0 ω') ∂μ| ≤ 1 := by diff --git a/Exchangeability/Probability/CenteredVariables.lean b/Exchangeability/Probability/CenteredVariables.lean index 4e080ebd..5603a35e 100644 --- a/Exchangeability/Probability/CenteredVariables.lean +++ b/Exchangeability/Probability/CenteredVariables.lean @@ -278,13 +278,11 @@ lemma correlation_coefficient_bounded (hZ_var_uniform : ∀ i, ∫ ω, (Z i ω)^2 ∂μ = ∫ ω, (Z 0 ω)^2 ∂μ) : -1 ≤ ρ ∧ ρ ≤ 1 := by -- Z 0 and Z 1 are in L²(μ) since they are bounded by M - have hZ0_L2 : MemLp (Z 0) 2 μ := by - apply memLp_two_of_bounded (hZ_meas 0) - exact hZ_bdd 0 + have hZ0_L2 : MemLp (Z 0) 2 μ := + MemLp.of_bound (hZ_meas 0).aestronglyMeasurable M (ae_of_all _ (hZ_bdd 0)) - have hZ1_L2 : MemLp (Z 1) 2 μ := by - apply memLp_two_of_bounded (hZ_meas 1) - exact hZ_bdd 1 + have hZ1_L2 : MemLp (Z 1) 2 μ := + MemLp.of_bound (hZ_meas 1).aestronglyMeasurable M (ae_of_all _ (hZ_bdd 1)) -- Apply Cauchy-Schwarz: |∫ Z₀·Z₁| ≤ sqrt(∫ Z₀²)·sqrt(∫ Z₁²) have h_CS := Exchangeability.Probability.IntegrationHelpers.abs_integral_mul_le_L2 hZ0_L2 hZ1_L2 diff --git a/Exchangeability/Probability/CondExp.lean b/Exchangeability/Probability/CondExp.lean index a4fd6549..4338fdea 100644 --- a/Exchangeability/Probability/CondExp.lean +++ b/Exchangeability/Probability/CondExp.lean @@ -394,16 +394,6 @@ section OperatorTheoretic variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} -/-- Bounded measurable functions are integrable on finite measures. - -Wraps `Integrable.of_bound` so callers can pass `⟨C, hC⟩` directly instead of -destructuring the bound first. -/ -lemma integrable_of_bounded [IsFiniteMeasure μ] - {f : Ω → ℝ} (hf : Measurable f) (hbd : ∃ C, ∀ ω, |f ω| ≤ C) : - Integrable f μ := by - obtain ⟨C, hC⟩ := hbd - exact ⟨hf.aestronglyMeasurable, HasFiniteIntegral.of_bounded (ae_of_all μ hC)⟩ - /-- **Conditional expectation is L¹-nonexpansive** (load-bearing lemma). For integrable functions f, g, the conditional expectation is contractive in L¹: diff --git a/Exchangeability/Probability/LpNormHelpers.lean b/Exchangeability/Probability/LpNormHelpers.lean index 293b3201..67930cc0 100644 --- a/Exchangeability/Probability/LpNormHelpers.lean +++ b/Exchangeability/Probability/LpNormHelpers.lean @@ -145,30 +145,6 @@ lemma eLpNorm_lt_of_integral_sq_lt /-! ### Membership in Lp Spaces -/ -/-- **Block average of bounded function is in L².** - -If `|f| ≤ M` everywhere on a probability space, then `f ∈ L²`. Used repeatedly in -Cesàro convergence proofs. -/ -lemma memLp_two_of_bounded - [IsProbabilityMeasure μ] {f : Ω → ℝ} {M : ℝ} - (hf_meas : Measurable f) - (hf_bdd : ∀ ω, |f ω| ≤ M) : - MemLp f 2 μ := by - refine MemLp.of_bound hf_meas.aestronglyMeasurable M ?_ - 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). -/ diff --git a/Exchangeability/Probability/SigmaAlgebraHelpers.lean b/Exchangeability/Probability/SigmaAlgebraHelpers.lean index eb13d522..8fe7ea5d 100644 --- a/Exchangeability/Probability/SigmaAlgebraHelpers.lean +++ b/Exchangeability/Probability/SigmaAlgebraHelpers.lean @@ -117,7 +117,11 @@ If `f n` are all Measurable[m] where `m ≤ m₀`, and `f n → g` a.e. (wrt a m then `g` is AEStronglyMeasurable[m] (with the witness being the limsup, which is Measurable[m]). This is the key lemma for "closedness" of L²[m] under L² limits: -we extract an a.e.-convergent subsequence and apply this. -/ +we extract an a.e.-convergent subsequence and apply this. + +Note: mathlib's `aestronglyMeasurable_of_tendsto_ae` covers only the `m = m₀` case (its +`AEStronglyMeasurable` is taken at the measure's own σ-algebra); this sub-σ-algebra version +has no mathlib equivalent as of v4.30.0. -/ lemma aestronglyMeasurable_sub_of_tendsto_ae {α : Type*} {m₀ : MeasurableSpace α} {μ : @MeasureTheory.Measure α m₀} {m : MeasurableSpace α} (_hm : m ≤ m₀) @@ -140,3 +144,4 @@ lemma aestronglyMeasurable_sub_of_tendsto_ae exact h_meas.stronglyMeasurable -- Conclude AEStronglyMeasurable[m] g using h as witness exact ⟨h, h_sm, h_ae_eq.symm⟩ + diff --git a/Exchangeability/Util/FinsetHelpers.lean b/Exchangeability/Util/FinsetHelpers.lean deleted file mode 100644 index 003a68aa..00000000 --- a/Exchangeability/Util/FinsetHelpers.lean +++ /dev/null @@ -1,49 +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.Data.Finset.Card -import Mathlib.Data.Fintype.Card - -/-! -# Finset Helper Lemmas - -General-purpose lemmas about Finsets and Fin types. - -## Main Results - -* `Finset.filter_val_lt_card`: Cardinality of filtered Fin elements by value bound --/ - -namespace Finset - -/-- **Cardinality of filtered Fin elements.** - -For `m ≥ n`, the number of elements `i : Fin m` with `i.val < n` is exactly `n`. - -This is because `Fin m = {0, 1, ..., m-1}` contains all of `{0, 1, ..., n-1}` when `m ≥ n`, -and these are precisely the elements satisfying `i.val < n`. - -The proof uses an explicit bijection between `Fin n` and the filtered set. -/ -lemma filter_val_lt_card {m n : ℕ} (h : m ≥ n) : - (Finset.filter (fun i : Fin m => i.val < n) Finset.univ).card = n := by - -- Establish bijection with Fin n via the natural inclusion - let f : Fin n → Fin m := Fin.castLE h - - have hf_inj : Function.Injective f := Fin.castLE_injective h - - have h_image : Finset.filter (fun i : Fin m => i.val < n) Finset.univ = - Finset.image f Finset.univ := by - ext i - simp only [mem_filter, mem_univ, true_and, mem_image] - constructor - · intro hi_lt - exact ⟨⟨i.val, hi_lt⟩, Fin.ext rfl⟩ - · rintro ⟨j, rfl⟩ - exact j.isLt - - rw [h_image, card_image_of_injective _ hf_inj] - simp only [card_univ, Fintype.card_fin] - -end Finset From ccc9c6c03f25af7d9dd2e881b669a9470e7d010e Mon Sep 17 00:00:00 2001 From: Cameron Freer Date: Wed, 10 Jun 2026 20:51:23 +0000 Subject: [PATCH 2/5] refactor: collapse injective_implies_strictMono_perm onto Tuple.sort; drop dead condExp_mul_pullout MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - injective_implies_strictMono_perm: 64-line orderIsoOfFin construction -> one-liner via Tuple.sort + Monotone.strictMono_of_injective (signature unchanged) - Probability/CondExp.lean condExp_mul_pullout: deleted — zero call sites; the same-named ViaKoopman/InfraGeneralized lemma (the one actually used, already a thin wrapper over mathlib's condExp_mul_of_aestronglyMeasurable_left) is unchanged --- Exchangeability/Probability/CondExp.lean | 38 ------------ Exchangeability/Util/StrictMono.lean | 73 ++---------------------- 2 files changed, 6 insertions(+), 105 deletions(-) diff --git a/Exchangeability/Probability/CondExp.lean b/Exchangeability/Probability/CondExp.lean index 4338fdea..73408188 100644 --- a/Exchangeability/Probability/CondExp.lean +++ b/Exchangeability/Probability/CondExp.lean @@ -417,44 +417,6 @@ lemma condExp_L1_lipschitz [IsFiniteMeasure μ] _ ≤ ∫ ω, |(f - g) ω| ∂μ := integral_abs_condExp_le (f - g) _ = ∫ ω, |f ω - g ω| ∂μ := rfl -/-- Conditional expectation pull-out property for bounded measurable functions. - -If g is m-measurable and bounded, then E[f·g|m] = E[f|m]·g a.e. -/ -lemma condExp_mul_pullout {Ω : Type*} {m₀ : MeasurableSpace Ω} {μ : Measure Ω} - [IsFiniteMeasure μ] - {m : MeasurableSpace Ω} (hm : m ≤ m₀) - {f g : Ω → ℝ} (hf : Integrable f μ) - (hg_meas : @Measurable Ω ℝ m _ g) - (hg_bd : ∃ C, ∀ ω, |g ω| ≤ C) : - μ[f * g|m] =ᵐ[μ] fun ω => μ[f|m] ω * g ω := by - -- Use mathlib's condExp_stronglyMeasurable_mul_of_bound with explicit instance management - -- following the pattern from condExpWith above. - - -- g is m-measurable, so it's m-strongly measurable - have hg_strong : StronglyMeasurable[m] g := hg_meas.stronglyMeasurable - - -- g is bounded - obtain ⟨C, hC⟩ := hg_bd - have hg_bound : ∀ᵐ ω ∂μ, ‖g ω‖ ≤ C := ae_of_all μ fun ω => (Real.norm_eq_abs _).le.trans (hC ω) - - -- Provide typeclass instances explicitly (IsFiniteMeasure is automatic via mathlib) - haveI : SigmaFinite (μ.trim hm) := inferInstance - - -- Now condExp_stronglyMeasurable_mul_of_bound can resolve instances - have h := condExp_stronglyMeasurable_mul_of_bound hm hg_strong hf C hg_bound - - -- Commute to get μ[f * g|m] = μ[f|m] * g - calc μ[f * g|m] - =ᵐ[μ] μ[g * f|m] := by - apply condExp_congr_ae - filter_upwards with ω - simp only [Pi.mul_apply] - ring - _ =ᵐ[μ] fun ω => g ω * μ[f|m] ω := h - _ =ᵐ[μ] fun ω => μ[f|m] ω * g ω := by - filter_upwards with ω - ring - end OperatorTheoretic end Exchangeability.Probability diff --git a/Exchangeability/Util/StrictMono.lean b/Exchangeability/Util/StrictMono.lean index d37a357f..f1a2a0d8 100644 --- a/Exchangeability/Util/StrictMono.lean +++ b/Exchangeability/Util/StrictMono.lean @@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Cameron Freer -/ import Mathlib.Data.Fin.Tuple.Basic +import Mathlib.Data.Fin.Tuple.Sort import Mathlib.Data.Finset.Sort /-! @@ -73,78 +74,16 @@ lemma strictMono_Fin_ge_id {k : Fin m → ℕ} (hk : StrictMono k) (i : Fin m) : /-- Any injective function `k : Fin m → ℕ` can be composed with a permutation to become strictly monotone. -**Construction:** Let `s := image k univ` (the image of k as a finset of ℕ). -Since k is injective, `s.card = m`. The `orderIsoOfFin` gives the sorted -enumeration of s. We define σ to map i to the position of k(i) in the sorted order. - -**Key property:** `(fun i => k (σ i))` is strictly increasing (sorted order). +The permutation is `Tuple.sort k`; monotone-after-sorting plus injectivity gives +strict monotonicity. This is a key lemma for reducing proofs about injective index selections to proofs about strictly monotone (consecutive-like) selections via contractability. -/ lemma injective_implies_strictMono_perm (k : Fin m → ℕ) (hk : Function.Injective k) : - ∃ (σ : Equiv.Perm (Fin m)), StrictMono (fun i => k (σ i)) := by - classical - -- Define the image of k as a finset - let s : Finset ℕ := Finset.image k Finset.univ - -- By injectivity, s has cardinality m - have hs : s.card = m := by - simp only [s, Finset.card_image_of_injective Finset.univ hk, Finset.card_univ, Fintype.card_fin] - -- Get the sorted enumeration of s - let sorted : Fin m ≃o ↑s := Finset.orderIsoOfFin s hs - -- For each i : Fin m, k(i) is in s, so we can find its sorted position - have hk_mem : ∀ i : Fin m, k i ∈ s := by - intro i - simp only [s, Finset.mem_image, Finset.mem_univ, true_and] - exact ⟨i, rfl⟩ - -- Define σ: for each position j in sorted order, find which i : Fin m maps to it - -- sorted j gives the j-th smallest element of s - -- We want σ such that k (σ j) = sorted j - -- Define σ⁻¹ first: σ⁻¹(i) = sorted position of k(i) - let σ_inv : Fin m → Fin m := fun i => - sorted.symm ⟨k i, hk_mem i⟩ - -- σ_inv is injective because sorted.symm and k are both injective - have hσ_inv_inj : Function.Injective σ_inv := by - intro i j hij - simp only [σ_inv] at hij - have h := sorted.symm.injective hij - simp only [Subtype.mk.injEq] at h - exact hk h - -- Since σ_inv : Fin m → Fin m is injective, it's a bijection - have hσ_inv_bij : Function.Bijective σ_inv := by - rw [Fintype.bijective_iff_injective_and_card] - exact ⟨hσ_inv_inj, rfl⟩ - -- Convert to an Equiv.Perm - let σ : Equiv.Perm (Fin m) := Equiv.ofBijective σ_inv hσ_inv_bij - -- Now σ.symm is the permutation we want - use σ.symm - -- Show k ∘ σ.symm is strictly monotone - intro i j hij - -- σ.symm(i) is the unique index such that σ_inv(σ.symm(i)) = i - -- i.e., sorted position of k(σ.symm(i)) is i - -- So k(σ.symm(i)) = sorted(i) (the i-th smallest element) - have h_eq_i : k (σ.symm i) = ↑(sorted i) := by - have h1 : σ_inv (σ.symm i) = i := by - simp only [σ, Equiv.ofBijective_apply_symm_apply] - simp only [σ_inv] at h1 - have h2 : sorted.symm ⟨k (σ.symm i), hk_mem (σ.symm i)⟩ = i := h1 - have h3 := sorted.apply_symm_apply ⟨k (σ.symm i), hk_mem (σ.symm i)⟩ - rw [h2] at h3 - exact Subtype.ext_iff.mp h3.symm - have h_eq_j : k (σ.symm j) = ↑(sorted j) := by - have h1 : σ_inv (σ.symm j) = j := by - simp only [σ, Equiv.ofBijective_apply_symm_apply] - simp only [σ_inv] at h1 - have h2 : sorted.symm ⟨k (σ.symm j), hk_mem (σ.symm j)⟩ = j := h1 - have h3 := sorted.apply_symm_apply ⟨k (σ.symm j), hk_mem (σ.symm j)⟩ - rw [h2] at h3 - exact Subtype.ext_iff.mp h3.symm - -- Goal: (fun i => k (σ.symm i)) i < (fun i => k (σ.symm i)) j - -- This simplifies to: k (σ.symm i) < k (σ.symm j) - simp only - rw [h_eq_i, h_eq_j] - -- sorted is an OrderIso, so it's strictly monotone - exact sorted.strictMono hij + ∃ (σ : Equiv.Perm (Fin m)), StrictMono (fun i => k (σ i)) := + ⟨Tuple.sort k, (Tuple.monotone_sort k).strictMono_of_injective + (hk.comp (Tuple.sort k).injective)⟩ end Exchangeability.Util.StrictMono From 30baa307b62730c7ce1f0f818b937eaab5a35bbb Mon Sep 17 00:00:00 2001 From: Cameron Freer Date: Wed, 10 Jun 2026 20:51:23 +0000 Subject: [PATCH 3/5] =?UTF-8?q?refactor!:=20eLpNorm=5Ftwo=5Fsq=5Feq=5Finte?= =?UTF-8?q?gral=5Fsq=20=E2=80=94=20shrink=20proof,=20drop=20IsFiniteMeasur?= =?UTF-8?q?e?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The new proof goes through MemLp.eLpNorm_eq_integral_rpow_norm and does not need [IsFiniteMeasure μ]; the hypothesis is dropped (user-approved signature weakening — strictly more general, call sites unaffected). ~55 lines of lintegral manipulation -> 8 lines. --- .../Probability/LpNormHelpers.lean | 65 ++----------------- 1 file changed, 7 insertions(+), 58 deletions(-) diff --git a/Exchangeability/Probability/LpNormHelpers.lean b/Exchangeability/Probability/LpNormHelpers.lean index 67930cc0..df03a880 100644 --- a/Exchangeability/Probability/LpNormHelpers.lean +++ b/Exchangeability/Probability/LpNormHelpers.lean @@ -47,69 +47,18 @@ the integral of f²: (eLpNorm f 2 μ)² = ∫ f² dμ This is a fundamental relationship used throughout probability theory, bridging -the gap between ENNReal-valued Lp norms and Real-valued integrals. - -**Proof strategy**: Use the definition of eLpNorm for p = 2, which involves -lintegral of ‖f‖^2, and convert via toReal. -/ +the gap between ENNReal-valued Lp norms and Real-valued integrals. -/ lemma eLpNorm_two_sq_eq_integral_sq - [IsFiniteMeasure μ] {f : Ω → ℝ} + {f : Ω → ℝ} (hf : MemLp f 2 μ) : (eLpNorm f 2 μ).toReal ^ 2 = ∫ ω, (f ω) ^ 2 ∂μ := by - -- Strategy: - -- 1. Use eLpNorm definition: eLpNorm f 2 μ = (∫⁻ ‖f‖²)^(1/2) - -- 2. Square both sides: (eLpNorm f 2 μ)² = ∫⁻ ‖f‖² - -- 3. Convert lintegral to integral: ∫⁻ ‖f‖² = ↑(∫ |f|²) = ↑(∫ f²) - - -- For real functions, ‖f‖² = |f|² = f² - have h_norm_eq : ∀ ω, ‖f ω‖ ^ 2 = (f ω) ^ 2 := - fun _ => by rw [Real.norm_eq_abs, sq_abs] - - -- Use the fundamental relationship for p = 2 - -- eLpNorm f p μ ^ p = ∫⁻ ‖f‖^p when p ≠ 0, ∞ - rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num : (2 : ℝ≥0∞) ≠ 0) - (by norm_num : (2 : ℝ≥0∞) ≠ ∞)] - - -- Simplify: ENNReal.toReal 2 = 2, so we have ((∫⁻ ‖f‖² )^(1/2)).toReal² - simp only [ENNReal.toReal_ofNat] - - -- Main strategy: Show (∫⁻ ‖f‖²).toReal = ∫ f² - -- Then use (a^(1/2))² = a to simplify the LHS - - -- Step 1: Rewrite LHS using ENNReal.toReal_rpow (backwards) - -- We have ((∫⁻ ...)^(1/2)).toReal ^ 2 and want (∫⁻ ...).toReal - conv_lhs => rw [← ENNReal.toReal_rpow] - - -- Step 2: Simplify (x^(1/2))^2 = x - rw [← Real.rpow_natCast _ 2, ← Real.rpow_mul ENNReal.toReal_nonneg] + have h0 : 0 ≤ ∫ ω, ‖f ω‖ ^ (2 : ℝ≥0∞).toReal ∂μ := + integral_nonneg fun ω => Real.rpow_nonneg (norm_nonneg _) _ + rw [hf.eLpNorm_eq_integral_rpow_norm (by norm_num) (by norm_num), + ENNReal.toReal_ofReal (Real.rpow_nonneg h0 _), + ← Real.rpow_natCast _ 2, ← Real.rpow_mul h0] norm_num - -- Step 3: Convert lintegral to integral for nonnegative functions - -- Key: ‖f ω‖ₑ = ↑‖f ω‖₊ where ‖·‖₊ is the nnnorm - -- First rewrite the lintegral in terms of ofReal - have h_enorm_conv : ∫⁻ (x : Ω), ‖f x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Ω), ENNReal.ofReal (‖f x‖ ^ 2) ∂μ := by - congr 1 - ext ω - -- Show ‖f ω‖ₑ ^ 2 = ENNReal.ofReal (‖f ω‖ ^ 2) - calc ‖f ω‖ₑ ^ 2 - = (↑‖f ω‖₊ : ℝ≥0∞) ^ 2 := by rw [enorm_eq_nnnorm] - _ = ↑(‖f ω‖₊ ^ 2) := by rw [← ENNReal.coe_pow] - _ = ENNReal.ofReal (↑(‖f ω‖₊ ^ 2) : ℝ) := by rw [ENNReal.ofReal_coe_nnreal] - _ = ENNReal.ofReal ((↑‖f ω‖₊ : ℝ) ^ 2) := by rw [NNReal.coe_pow] - _ = ENNReal.ofReal (‖f ω‖ ^ 2) := by rw [coe_nnnorm] - rw [h_enorm_conv] - -- Now use the fundamental relationship: (∫⁻ ofReal g).toReal = ∫ g for nonnegative g - rw [← integral_eq_lintegral_of_nonneg_ae] - · congr 1 - ext ω - exact h_norm_eq ω - · -- Nonnegativity: ‖f ω‖ ^ 2 ≥ 0 - apply ae_of_all - intro ω - exact sq_nonneg _ - · -- AE measurability - apply AEStronglyMeasurable.pow - exact hf.1.norm - /-- **L² norm bound from integral bound.** If the integral of f² is less than r², then the L² norm of f is less than r. From 4a248d8eb7e9bf235854f7fac8469594a4dddefe Mon Sep 17 00:00:00 2001 From: Cameron Freer Date: Wed, 10 Jun 2026 20:59:42 +0000 Subject: [PATCH 4/5] chore(artifacts): regenerate import graph (103 -> 102 modules) + refresh counts Util/FinsetHelpers.lean deletion drops one module: SVG/PNG/HTML regenerated via the PR #132 procedure (colors carried per module; 102 clickable SVG nodes; embedded GEXF re-validated, 102 nodes / 151 edges); home_page file count and STATUS.md LOC table refreshed. --- STATUS.md | 8 +- blueprint/web/import_graph_colored.html | 2 +- blueprint/web/import_graph_colored.png | Bin 308031 -> 286305 bytes blueprint/web/import_graph_colored.svg | 1344 +++++++++++------------ home_page/index.md | 2 +- 5 files changed, 671 insertions(+), 685 deletions(-) diff --git a/STATUS.md b/STATUS.md index f6f93b46..155a8468 100644 --- a/STATUS.md +++ b/STATUS.md @@ -1,6 +1,6 @@ # Project Status: de Finetti Theorem Formalization -**Last Updated:** 2026-06-09 +**Last Updated:** 2026-06-10 ## Executive Summary @@ -11,9 +11,9 @@ LOC counts below cover the `Exchangeability/` library tree (excluding the top-le |---------------|------:|-------|--------| | **ViaMartingale** | 2,804 | Pass | **COMPLETE** | | **ViaL2** | 7,443 | Pass | **COMPLETE** | -| **ViaKoopman** | 5,305 | Pass | **COMPLETE** | -| *Shared infrastructure* | 11,111 | — | — | -| **Total** | **26,663** | Pass | **COMPLETE** | +| **ViaKoopman** | 5,314 | Pass | **COMPLETE** | +| *Shared infrastructure* | 10,881 | — | — | +| **Total** | **26,442** | Pass | **COMPLETE** | **All three proofs complete.** diff --git a/blueprint/web/import_graph_colored.html b/blueprint/web/import_graph_colored.html index f5ce0a05..b6dde420 100644 --- a/blueprint/web/import_graph_colored.html +++ b/blueprint/web/import_graph_colored.html @@ -501,7 +501,7 @@

Import Graph for Exchangeability

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+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 - - + + 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+ + 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.IntegrationHelpers - -Exchangeability.Probability.IntegrationHelpers + +Exchangeability.Probability.IntegrationHelpers Exchangeability.Probability.IntegrationHelpers->Exchangeability.Probability.Martingale.Crossings.AntitoneLimit - - + + Exchangeability.Probability.CenteredVariables - -Exchangeability.Probability.CenteredVariables + +Exchangeability.Probability.CenteredVariables - + Exchangeability.Probability.IntegrationHelpers->Exchangeability.Probability.CenteredVariables - - + + - + Exchangeability.Probability.CondExp - -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.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.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.DeFinetti.ViaKoopman.Infrastructure - -Exchangeability.DeFinetti.ViaKoopman.Infrastructure + +Exchangeability.DeFinetti.ViaKoopman.Infrastructure - + Exchangeability.DeFinetti.ViaKoopman.Infrastructure->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 cdd67dec..c08e6223 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) | 103 files, clickable links to docs | +| [Modules (SVG)]({{ site.url }}/blueprint/import_graph_colored.svg) | 102 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 1523ebf736e941b2ed87c34941540b9d4367ce1e Mon Sep 17 00:00:00 2001 From: Cameron Freer Date: Wed, 10 Jun 2026 21:10:54 +0000 Subject: [PATCH 5/5] style: drop blank line at EOF in SigmaAlgebraHelpers MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Also recording here (per review): commit ccc9c6c0's removal of the public lemma Exchangeability.Probability.condExp_mul_pullout is an intentional breaking API change — it had zero call sites in this repo and no wrapper is kept; external users should migrate to mathlib's condExp_mul_of_stronglyMeasurable_right (right-factor form) or the ViaKoopman condExp_mul_pullout (left-factor, shift-invariant form). --- Exchangeability/Probability/SigmaAlgebraHelpers.lean | 1 - 1 file changed, 1 deletion(-) diff --git a/Exchangeability/Probability/SigmaAlgebraHelpers.lean b/Exchangeability/Probability/SigmaAlgebraHelpers.lean index 8fe7ea5d..3dcdb259 100644 --- a/Exchangeability/Probability/SigmaAlgebraHelpers.lean +++ b/Exchangeability/Probability/SigmaAlgebraHelpers.lean @@ -144,4 +144,3 @@ lemma aestronglyMeasurable_sub_of_tendsto_ae exact h_meas.stronglyMeasurable -- Conclude AEStronglyMeasurable[m] g using h as witness exact ⟨h, h_sm, h_ae_eq.symm⟩ -