openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Recovery /InheritedMaskMargins.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMaskConcentration | |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMasks | |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedStatisticWindows | |
| namespace OAI | |
| /-! Finite orbit symmetries, masks and exact recovery operations. -/ | |
| namespace MatrixMultiplication.PermutationMatching | |
| open scoped BigOperators | |
| open MatrixMultiplication.InheritedMasks | |
| open Classical | |
| noncomputable section | |
| section ClassMixtures | |
| variable {C : Type*} [Fintype C] [DecidableEq C] {P : C → Type*} | |
| [∀ c, Fintype (P c)] [∀ c, DecidableEq (P c)] | |
| def classWeight (P : C → Type*) [∀ c, Fintype (P c)] (c : C) : ℝ := | |
| (Fintype.card (P c) : ℝ) / populationSize P | |
| def classDensity (A : ∀ c, Finset (P c)) (c : C) : ℝ := | |
| ((A c).card : ℝ) / Fintype.card (P c) | |
| def classProductMixture (P : C → Type*) [∀ c, Fintype (P c)] | |
| (left right : C → ℝ) : ℝ := | |
| ∑ c, classWeight P c * (left c * right c) | |
| omit [DecidableEq C] [∀ c, DecidableEq (P c)] in | |
| theorem classWeight_nonneg (hm : 0 < populationSize P) (c : C) : | |
| 0 ≤ classWeight P c := div_nonneg (Nat.cast_nonneg _) (le_of_lt hm) | |
| omit [DecidableEq C] [∀ c, DecidableEq (P c)] in | |
| theorem sum_classWeight (hm : 0 < populationSize P) : | |
| ∑ c, classWeight P c = 1 := by | |
| unfold classWeight | |
| rw [← Finset.sum_div] | |
| exact div_self (ne_of_gt hm) | |
| omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (P c)] in | |
| theorem classDensity_nonneg (A : ∀ c, Finset (P c)) (c : C) : | |
| 0 ≤ classDensity A c := div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _) | |
| omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (P c)] in | |
| theorem classDensity_le_one [∀ c, Nonempty (P c)] | |
| (A : ∀ c, Finset (P c)) (c : C) : classDensity A c ≤ 1 := by | |
| have hn : (0 : ℝ) < Fintype.card (P c) := by exact_mod_cast Fintype.card_pos | |
| apply (div_le_iff₀ hn).2 | |
| simpa using (Nat.cast_le.mpr (Finset.card_le_univ (A c)) : | |
| ((A c).card : ℝ) ≤ Fintype.card (P c)) | |
| omit [DecidableEq C] in | |
| theorem totalMatchingMean_normalized_eq [∀ c, Nonempty (P c)] | |
| (A B : ∀ c, Finset (P c)) (hm : 0 < populationSize P) : | |
| totalMatchingMean A B / populationSize P = | |
| classProductMixture P (classDensity A) (classDensity B) := by | |
| classical | |
| rw [totalMatchingMean_eq, Finset.sum_div] | |
| unfold classProductMixture | |
| apply Finset.sum_congr rfl | |
| intro c _ | |
| have hn : (Fintype.card (P c) : ℝ) ≠ 0 := by | |
| exact_mod_cast (ne_of_gt (Fintype.card_pos (α := P c))) | |
| dsimp [classWeight, classDensity] | |
| field_simp [hn, ne_of_gt hm] | |
| omit [DecidableEq C] in | |
| theorem totalMatchingMean_prescribed_error [∀ c, Nonempty (P c)] | |
| (A B : ∀ c, Finset (P c)) (hm : 0 < populationSize P) | |
| (left right : C → ℝ) (τ : ℝ) | |
| (hright0 : ∀ c, 0 ≤ right c) (hright1 : ∀ c, right c ≤ 1) | |
| (hleft : ∀ c, |classDensity A c - left c| ≤ τ) | |
| (hright : ∀ c, |classDensity B c - right c| ≤ τ) : | |
| |totalMatchingMean A B / populationSize P - | |
| classProductMixture P left right| ≤ 2 * τ := by | |
| rw [totalMatchingMean_normalized_eq A B hm] | |
| have h := weighted_window (classWeight P) | |
| (fun c => classDensity A c * classDensity B c - left c * right c) | |
| 0 (2 * τ) (classWeight_nonneg hm) (sum_classWeight hm) (by | |
| intro c | |
| simpa only [sub_zero] using product_coordinate_error | |
| (classDensity_nonneg A c) (classDensity_le_one A c) | |
| (hright0 c) (hright1 c) (hleft c) (hright c)) | |
| simpa only [classProductMixture, sub_zero, mul_sub, Finset.sum_sub_distrib] using h | |
| omit [DecidableEq C] in | |
| theorem totalMatchingMean_inherited_margin [∀ c, Nonempty (P c)] | |
| (A B : ∀ c, Finset (P c)) (hm : 0 < populationSize P) | |
| (left right : C → ℝ) (η τ α ν : ℝ) | |
| (hright0 : ∀ c, 0 ≤ right c) (hright1 : ∀ c, right c ≤ 1) | |
| (hleft : ∀ c, |classDensity A c - left c| ≤ τ) | |
| (hright : ∀ c, |classDensity B c - right c| ≤ τ) | |
| (happrox : |classProductMixture P left right - ν| ≤ α) | |
| (hτ : τ ≤ η / 8) (hα : α ≤ η / 4) : | |
| |totalMatchingMean A B / populationSize P - ν| ≤ η / 2 := by | |
| have hchild := totalMatchingMean_prescribed_error A B hm left right τ | |
| hright0 hright1 hleft hright | |
| have h := inherited_margin (actual := ν) | |
| (expected := classProductMixture P left right) | |
| (prescribed := totalMatchingMean A B / populationSize P) | |
| (by simpa only [abs_sub_comm] using hchild) | |
| (by simpa only [abs_sub_comm] using happrox) hτ hα | |
| simpa only [abs_sub_comm] using h | |
| theorem inheritedMask_rejection_of_windows_le [∀ c, Nonempty (P c)] | |
| (A B : ∀ c, Finset (P c)) (hn : ∀ c, 2 ≤ Fintype.card (P c)) | |
| (hm : 0 < populationSize P) (left right : C → ℝ) (η τ α ν : ℝ) | |
| (hη : 0 < η) | |
| (hright0 : ∀ c, 0 ≤ right c) (hright1 : ∀ c, right c ≤ 1) | |
| (hleft : ∀ c, |classDensity A c - left c| ≤ τ) | |
| (hright : ∀ c, |classDensity B c - right c| ≤ τ) | |
| (happrox : |classProductMixture P left right - ν| ≤ α) | |
| (hτ : τ ≤ η / 8) (hα : α ≤ η / 4) : | |
| average (fun p : ClassPermutations P => | |
| if η < |totalMatchingCount A B p / populationSize P - ν| then 1 else 0) ≤ | |
| 1 / (2 * populationSize P * η ^ 2) := by | |
| exact inheritedMask_rejection_le A B hn hm η ν hη | |
| (totalMatchingMean_inherited_margin A B hm left right η τ α ν | |
| hright0 hright1 hleft hright happrox hτ hα) | |
| theorem twoHalf_inheritedMask_rejection_of_windows_le [∀ c, Nonempty (P c)] | |
| (A B : ∀ c, Finset (P c)) (hn : ∀ c, 2 ≤ Fintype.card (P c)) | |
| (hm : 0 < populationSize P) (left right : C → ℝ) (η τ α ν : ℝ) | |
| (hη : 0 < η) | |
| (hright0 : ∀ c, 0 ≤ right c) (hright1 : ∀ c, right c ≤ 1) | |
| (hleft : ∀ c, |classDensity A c - left c| ≤ τ) | |
| (hright : ∀ c, |classDensity B c - right c| ≤ τ) | |
| (happrox : |classProductMixture P left right - ν| ≤ α) | |
| (hτ : τ ≤ η / 8) (hα : α ≤ η / 4) : | |
| average (fun p : ClassPermutations P × ClassPermutations P => | |
| if η < |totalTwoHalfCount A B p / populationSize P - ν| then 1 else 0) ≤ | |
| 1 / (2 * populationSize P * η ^ 2) := by | |
| exact twoHalf_inheritedMask_rejection_le A B hn hm η ν hη | |
| (totalMatchingMean_inherited_margin A B hm left right η τ α ν | |
| hright0 hright1 hleft hright happrox hτ hα) | |
| theorem selectedTwoHalfMask_rejection_of_windows_le | |
| (D : Finset C) (A B : ∀ c, Finset (P c)) | |
| (hn : ∀ c ∈ D, 2 ≤ Fintype.card (P c)) | |
| (hm : 0 < populationSize (fun c : D => P c)) | |
| (left right : C → ℝ) (η τ α ν : ℝ) (hη : 0 < η) | |
| (hright0 : ∀ c ∈ D, 0 ≤ right c) (hright1 : ∀ c ∈ D, right c ≤ 1) | |
| (hleft : ∀ c ∈ D, |classDensity A c - left c| ≤ τ) | |
| (hright : ∀ c ∈ D, |classDensity B c - right c| ≤ τ) | |
| (happrox : |classProductMixture (fun c : D => P c) | |
| (fun c => left c) (fun c => right c) - ν| ≤ α) | |
| (hτ : τ ≤ η / 8) (hα : α ≤ η / 4) : | |
| average (fun p : ClassPermutations P × ClassPermutations P => | |
| if η < |selectedTwoHalfCount D A B p / | |
| populationSize (fun c : D => P c) - ν| then 1 else 0) ≤ | |
| 1 / (2 * populationSize (fun c : D => P c) * η ^ 2) := by | |
| let : ∀ c : D, Nonempty (P c) := fun c => | |
| Fintype.card_pos_iff.mp (lt_of_lt_of_le (by decide : 0 < 2) (hn c c.property)) | |
| apply selectedTwoHalfMask_rejection_le D A B hn hm η ν hη | |
| exact totalMatchingMean_inherited_margin (fun c : D => A c) (fun c : D => B c) | |
| hm (fun c => left c) (fun c => right c) η τ α ν | |
| (fun c => hright0 c c.property) (fun c => hright1 c c.property) | |
| (fun c => hleft c c.property) (fun c => hright c c.property) happrox hτ hα | |
| end ClassMixtures | |
| section CompleteWordStatistics | |
| variable {C : Type*} [Fintype C] [DecidableEq C] {P X Y S T : C → Type*} | |
| [∀ c, Fintype (P c)] [∀ c, DecidableEq (P c)] | |
| [∀ c, Fintype (S c)] [∀ c, DecidableEq (S c)] | |
| [∀ c, Fintype (T c)] [∀ c, DecidableEq (T c)] | |
| omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (P c)] in | |
| theorem classDensity_statisticPositions (w : ∀ c, P c → X c) | |
| (statistic : ∀ c, X c → S c) (s : ∀ c, S c) (c : C) : | |
| classDensity (fun c => statisticPositions (w c) (statistic c) (s c)) c = | |
| empiricalLaw (statistic c ∘ w c) (s c) := by | |
| rw [classDensity, statisticPositions_card, empiricalLaw] | |
| omit [DecidableEq C] in | |
| theorem totalMatchingMean_statistic_eq [∀ c, Nonempty (P c)] | |
| (w : ∀ c, P c → X c) (v : ∀ c, P c → Y c) | |
| (leftStat : ∀ c, X c → S c) (rightStat : ∀ c, Y c → T c) | |
| (s : ∀ c, S c) (t : ∀ c, T c) (hm : 0 < populationSize P) : | |
| totalMatchingMean (fun c => statisticPositions (w c) (leftStat c) (s c)) | |
| (fun c => statisticPositions (v c) (rightStat c) (t c)) / populationSize P = | |
| classProductMixture P | |
| (fun c => empiricalLaw (leftStat c ∘ w c) (s c)) | |
| (fun c => empiricalLaw (rightStat c ∘ v c) (t c)) := by | |
| rw [totalMatchingMean_normalized_eq _ _ hm] | |
| simp only [classProductMixture, classDensity_statisticPositions] | |
| theorem twoHalf_statistic_rejection_of_typeWindows_le [∀ c, Nonempty (P c)] | |
| (w : ∀ c, P c → X c) (v : ∀ c, P c → Y c) | |
| (leftStat : ∀ c, X c → S c) (rightStat : ∀ c, Y c → T c) | |
| (s : ∀ c, S c) (t : ∀ c, T c) | |
| (hn : ∀ c, 2 ≤ Fintype.card (P c)) (hm : 0 < populationSize P) | |
| (leftLaw : ∀ c, S c → ℝ) (rightLaw : ∀ c, T c → ℝ) | |
| (η τ α ν : ℝ) (hη : 0 < η) | |
| (hright0 : ∀ c, 0 ≤ rightLaw c (t c)) | |
| (hright1 : ∀ c, rightLaw c (t c) ≤ 1) | |
| (hleft : ∀ c, typeWindow (leftLaw c) τ (leftStat c ∘ w c)) | |
| (hright : ∀ c, typeWindow (rightLaw c) τ (rightStat c ∘ v c)) | |
| (happrox : |classProductMixture P | |
| (fun c => leftLaw c (s c)) (fun c => rightLaw c (t c)) - ν| ≤ α) | |
| (hτ : τ ≤ η / 8) (hα : α ≤ η / 4) : | |
| average (fun p : ClassPermutations P × ClassPermutations P => | |
| if η < |totalTwoHalfCount | |
| (fun c => statisticPositions (w c) (leftStat c) (s c)) | |
| (fun c => statisticPositions (v c) (rightStat c) (t c)) p / | |
| populationSize P - ν| then 1 else 0) ≤ | |
| 1 / (2 * populationSize P * η ^ 2) := by | |
| apply twoHalf_inheritedMask_rejection_of_windows_le _ _ hn hm | |
| (fun c => leftLaw c (s c)) (fun c => rightLaw c (t c)) η τ α ν hη | |
| hright0 hright1 | |
| · intro c | |
| simpa only [classDensity_statisticPositions] using hleft c (s c) | |
| · intro c | |
| simpa only [classDensity_statisticPositions] using hright c (t c) | |
| · exact happrox | |
| · exact hτ | |
| · exact hα | |
| theorem selectedTwoHalf_statistic_rejection_of_completeWindows_le | |
| [∀ c, Fintype (X c)] [∀ c, DecidableEq (X c)] | |
| [∀ c, Fintype (Y c)] [∀ c, DecidableEq (Y c)] | |
| (D : Finset C) (w : ∀ c, P c → X c) (v : ∀ c, P c → Y c) | |
| (leftStat : ∀ c, X c → S c) (rightStat : ∀ c, Y c → T c) | |
| (s : ∀ c, S c) (t : ∀ c, T c) | |
| (hn : ∀ c ∈ D, 2 ≤ Fintype.card (P c)) | |
| (hm : 0 < populationSize (fun c : D => P c)) | |
| (leftLaw : ∀ c, X c → ℝ) (rightLaw : ∀ c, Y c → ℝ) | |
| (leftε rightε : C → ℝ) (η α ν : ℝ) (hη : 0 < η) | |
| (hright0 : ∀ c ∈ D, ∀ y, 0 ≤ rightLaw c y) | |
| (hrightSum : ∀ c ∈ D, ∑ y, rightLaw c y = 1) | |
| (hleftε : ∀ c ∈ D, 0 ≤ leftε c) (hrightε : ∀ c ∈ D, 0 ≤ rightε c) | |
| (hleft : ∀ c ∈ D, typeWindow (leftLaw c) (leftε c) (w c)) | |
| (hright : ∀ c ∈ D, typeWindow (rightLaw c) (rightε c) (v c)) | |
| (hleftFactor : ∀ c ∈ D, (Fintype.card (X c) : ℝ) * leftε c ≤ η / 8) | |
| (hrightFactor : ∀ c ∈ D, (Fintype.card (Y c) : ℝ) * rightε c ≤ η / 8) | |
| (happrox : |classProductMixture (fun c : D => P c) | |
| (fun c => pushforwardLaw (leftStat c) (leftLaw c) (s c)) | |
| (fun c => pushforwardLaw (rightStat c) (rightLaw c) (t c)) - ν| ≤ α) | |
| (hα : α ≤ η / 4) : | |
| average (fun p : ClassPermutations P × ClassPermutations P => | |
| if η < |selectedTwoHalfCount D | |
| (fun c => statisticPositions (w c) (leftStat c) (s c)) | |
| (fun c => statisticPositions (v c) (rightStat c) (t c)) p / | |
| populationSize (fun c : D => P c) - ν| then 1 else 0) ≤ | |
| 1 / (2 * populationSize (fun c : D => P c) * η ^ 2) := by | |
| apply selectedTwoHalfMask_rejection_of_windows_le D _ _ hn hm | |
| (fun c => pushforwardLaw (leftStat c) (leftLaw c) (s c)) | |
| (fun c => pushforwardLaw (rightStat c) (rightLaw c) (t c)) | |
| η (η / 8) α ν hη | |
| · intro c hc | |
| exact pushforwardLaw_nonneg (rightStat c) (rightLaw c) (hright0 c hc) (t c) | |
| · intro c hc | |
| exact pushforwardLaw_le_one (rightStat c) (rightLaw c) | |
| (hright0 c hc) (hrightSum c hc) (t c) | |
| · intro c hc | |
| rw [classDensity_statisticPositions] | |
| exact (typeWindow_statistic (leftLaw c) (leftε c) (w c) (leftStat c) | |
| (hleftε c hc) (hleft c hc) (s c)).trans (hleftFactor c hc) | |
| · intro c hc | |
| rw [classDensity_statisticPositions] | |
| exact (typeWindow_statistic (rightLaw c) (rightε c) (v c) (rightStat c) | |
| (hrightε c hc) (hright c hc) (t c)).trans (hrightFactor c hc) | |
| · exact happrox | |
| · exact le_rfl | |
| · exact hα | |
| end CompleteWordStatistics | |
| end | |
| end MatrixMultiplication.PermutationMatching | |
| end OAI | |