openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldHistory /IncomingMasks.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.IncomingCore | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.MaskLaws | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.RawMasks | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.StageCIncoming | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Tensors | |
| namespace OAI | |
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldHistoryIncomingMasks | |
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldParameters | |
| open AllFieldHistoryChildLaws AllFieldHistorySupport AllFieldHistoryMasks | |
| open AllFieldHistoryMaskLaws AllFieldHistoryRawMasks InheritedMasks | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K tick : ℕ} | |
| theorem empiricalLaw_congr_decidable {P A : Type*} [Fintype P] | |
| (d₁ d₂ : DecidableEq A) (w : P → A) (a : A) : | |
| @empiricalLaw P A _ d₁ w a = @empiricalLaw P A _ d₂ w a := by | |
| cases Subsingleton.elim d₁ d₂ | |
| rfl | |
| theorem typeWindow_mono_decidable {P A : Type*} [Fintype P] | |
| (d₁ d₂ : DecidableEq A) (ν : A → ℝ) {η δ : ℝ} (hη : η ≤ δ) | |
| (w : P → A) (hw : @typeWindow P A _ d₁ ν η w) : | |
| @typeWindow P A _ d₂ ν δ w := by | |
| cases Subsingleton.elim d₁ d₂ | |
| intro a | |
| exact (hw a).trans hη | |
| theorem weight_append {m n : ℕ} (l : Fin m → Fin 7) (r : Fin n → Fin 7) : | |
| CWStrands.weight (Fin.append l r) = CWStrands.weight l + CWStrands.weight r := by | |
| simp [CWStrands.weight, Fin.sum_univ_add] | |
| theorem strand_append {F : Type*} [CommRing F] {m n : ℕ} | |
| (x y z : (Fin m → Fin 7) × (Fin n → Fin 7)) : | |
| CWStrands.strand (F := F) (Fin (m + n)) | |
| (Fin.append x.1 x.2) (Fin.append y.1 y.2) (Fin.append z.1 z.2) = | |
| CWStrands.strand (Fin m) x.1 y.1 z.1 * CWStrands.strand (Fin n) x.2 y.2 z.2 := by | |
| simp [CWStrands.strand, Fin.prod_univ_add] | |
| theorem shapeTensor_append {F : Type*} [CommRing F] {m n : ℕ} (g : Shape) | |
| (x y z : (Fin m → Fin 7) × (Fin n → Fin 7)) : | |
| CWStrands.shapeTensor (F := F) (Fin (m + n)) g | |
| (Fin.append x.1 x.2) (Fin.append y.1 y.2) (Fin.append z.1 z.2) = | |
| CWStrands.shapeTensor (Fin m ⊕ Fin n) g | |
| (Sum.elim x.1 x.2) (Sum.elim y.1 y.2) (Sum.elim z.1 z.2) := by | |
| simp only [CWStrands.shapeTensor, weight_append, CWStrands.weight_sum_elim, | |
| strand_append, CWStrands.strand_sum_elim, Tensor.product] | |
| theorem shapeTensor_joinHalves {F : Type*} [CommRing F] (w : Work K) (g : Shape) | |
| (x y z : (Fin w.halfLength → Fin 7) × (Fin w.halfLength → Fin 7)) : | |
| CWStrands.shapeTensor (F := F) (Fin (currentLength w.source)) g | |
| (joinHalves w x) (joinHalves w y) (joinHalves w z) = | |
| CWStrands.shapeTensor (Fin w.halfLength ⊕ Fin w.halfLength) g | |
| (Sum.elim x.1 x.2) (Sum.elim y.1 y.2) (Sum.elim z.1 z.2) := by | |
| cases w <;> exact shapeTensor_append g x y z | |
| def defaultStatistic : (w : Work K) → w.Statistic | |
| | .stageA _ => (0, 0) | |
| | .stageB _ => (0 : Fin 6) | |
| | .stageC _ => PUnit.unit | |
| def choiceAt (h : Active K tick) (a : Statistic h) : SymbolChoice K tick := | |
| Function.update (fun g => defaultStatistic g.val.1) h a | |
| @[simp] theorem choiceAt_self (h : Active K tick) (a : Statistic h) : | |
| choiceAt h a h = a := by simp [choiceAt] | |
| theorem rawPairCount_eq_wordPopulation (allocation : Allocation) (m : ℕ) | |
| (i : MaskIndex K tick) (w : ActiveRawPairs (K := K) (tick := tick) allocation m) : | |
| rawPairCount allocation m i w = | |
| (wordPopulation (fun j => (statistic i.1 (w i.1 j).1, | |
| statistic i.1 (w i.1 j).2)) (i.2.1 i.1, i.2.2 i.1) : ℝ) := by | |
| simp only [rawPairCount, wordPopulation, Fintype.card_subtype, | |
| Finset.natCast_card_filter, Prod.mk.injEq] | |
| @[simp] theorem fourStatistic_append (l r : Fin 2 → Fin 7) : | |
| CWCompleteStatistics.fourStatistic (Fin.append l r) = | |
| (CWCompleteStatistics.twoStatistic l, CWCompleteStatistics.twoStatistic r) := rfl | |
| @[simp] theorem append_singletons (l r : Fin 1 → Fin 7) : | |
| Fin.append l r = ![l 0, r 0] := by | |
| funext i | |
| fin_cases i <;> rfl | |
| theorem rawPass_stageB_typeWindow (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (side : Fin 3) (w : ActiveRawPairs (K := K) (tick := tick) allocation m) | |
| (hw : rawPass allocation m ε side w) (h : APositive K) (phi : Placement) | |
| (ht : FiniteSchedule.lotTick (Work.stageB h).lot (Work.stageB h).stage = tick) : | |
| typeWindow (fun ij : PairSlot => (halfLaw (aShape h.val) (phi.symm side) ij : ℝ)) | |
| (pairWidth ε (stageBActive h phi ht)) | |
| (fun j => CWCompleteStatistics.fourStatistic | |
| (joinHalves (.stageB h) (w (stageBActive h phi ht) j))) := by | |
| intro ij | |
| have hp := hw (stageBActive h phi ht, | |
| choiceAt (stageBActive h phi ht) ij.1, choiceAt (stageBActive h phi ht) ij.2) | |
| rw [center_stageB] at hp | |
| simp only [choiceAt_self (stageBActive h phi ht) ij.1, | |
| choiceAt_self (stageBActive h phi ht) ij.2] at hp | |
| have hcard : Fintype.card (JointPopulation.Positions (activeCounts allocation m) | |
| (stageBActive h phi ht)) = population allocation m | |
| ((stageBActive h phi ht).val.1.source, (stageBActive h phi ht).val.2) := by | |
| exact (Fintype.card_congr | |
| (sourcePositionsEquiv allocation m (stageBActive h phi ht))).trans | |
| (Fintype.card_fin _) | |
| have hcount : (wordPopulation (fun j => | |
| (statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).1, | |
| statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).2)) ij : ℝ) = | |
| rawPairCount allocation m (stageBActive h phi ht, | |
| choiceAt (stageBActive h phi ht) ij.1, choiceAt (stageBActive h phi ht) ij.2) w := by | |
| rcases ij with ⟨a, b⟩ | |
| simp only [wordPopulation, Fintype.card_subtype, | |
| Finset.natCast_card_filter, rawPairCount] | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| apply if_congr | |
| · simp only [choiceAt_self (stageBActive h phi ht) a, | |
| choiceAt_self (stageBActive h phi ht) b] | |
| exact Prod.mk_inj | |
| · rfl | |
| · rfl | |
| have hword : (fun j => CWCompleteStatistics.fourStatistic | |
| (joinHalves (.stageB h) (w (stageBActive h phi ht) j))) = | |
| (fun j => (statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).1, | |
| statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).2)) := by | |
| funext j | |
| rfl | |
| have he : empiricalLaw (fun j => CWCompleteStatistics.fourStatistic | |
| (joinHalves (.stageB h) (w (stageBActive h phi ht) j))) ij = | |
| empiricalLaw (fun j => | |
| (statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).1, | |
| statistic (stageBActive h phi ht) (w (stageBActive h phi ht) j).2)) ij := by | |
| rw [hword] | |
| exact empiricalLaw_congr_decidable _ _ _ _ | |
| rw [he] | |
| unfold empiricalLaw | |
| rw [hcard, hcount] | |
| exact hp | |
| attribute [local irreducible] activeCounts population | |
| theorem incoming_stageA (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (side : Fin 3) (w : ActiveRawPairs (K := K) (tick := tick) allocation m) | |
| (h : InitialPositive K) (phi : Placement) | |
| (ht : FiniteSchedule.lotTick (Work.stageA h).lot (Work.stageA h).stage = tick) : | |
| residentMask allocation m ε ((Work.stageA h).source, phi) side | |
| (incomingWord allocation m ⟨(.stageA h, phi), ht⟩ w) := by | |
| trivial | |
| theorem incoming_stageB (allocation : Allocation) (m : ℕ) {ε : ℝ} (hε : 0 ≤ ε) | |
| (side : Fin 3) (w : ActiveRawPairs (K := K) (tick := tick) allocation m) | |
| (hw : rawPass allocation m ε side w) (h : APositive K) (phi : Placement) | |
| (ht : FiniteSchedule.lotTick (Work.stageB h).lot (Work.stageB h).stage = tick) : | |
| residentMask allocation m ε ((Work.stageB h).source, phi) side | |
| (incomingWord allocation m (stageBActive h phi ht) w) := by | |
| intro _ | |
| have hraw := rawPass_stageB_typeWindow allocation m ε side w hw h phi ht | |
| have htransport := (JointCanonicalization.typeWindow_reindex | |
| (fun ij : PairSlot => (halfLaw (aShape h.val) (phi.symm side) ij : ℝ)) | |
| (pairWidth ε (stageBActive h phi ht)) | |
| (fun j => CWCompleteStatistics.fourStatistic | |
| (joinHalves (.stageB h) (w (stageBActive h phi ht) j))) | |
| (sourcePositionsEquiv allocation m (stageBActive h phi ht)).symm).2 hraw | |
| have hwidth : pairWidth ε (stageBActive h phi ht) ≤ ε / 8 := by | |
| change ε / 16 ^ 1 ≤ ε / 8 | |
| norm_num | |
| linarith | |
| dsimp only [Work.source, residentLaw, residentLawRat, residentWidth, residentStatistic] | |
| exact typeWindow_mono_decidable _ _ _ hwidth _ htransport | |
| theorem incoming_stageC (allocation : Allocation) {m : ℕ} (hm : 0 < m) | |
| {ε : ℝ} (hε : 0 ≤ ε) (side : Fin 3) | |
| (e : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m)) | |
| (w : ActiveRawPairs (K := K) (tick := tick) allocation m) | |
| (h : PartC K) (phi : Placement) | |
| (ht : FiniteSchedule.lotTick (Work.stageC h).lot (Work.stageC h).stage = tick) | |
| (hweights : ∀ j, | |
| CWLeafStatistics.weight ((w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).1 (0 : Fin 1)) = | |
| (JointPopulation.shapeSide side ((e (AllFieldHistoryStageCIncoming.stageCActive h phi ht)).val j)).val ∧ | |
| CWLeafStatistics.weight ((w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).2 (0 : Fin 1)) = | |
| activeParentShape (AllFieldHistoryStageCIncoming.stageCActive h phi ht) side - | |
| (JointPopulation.shapeSide side ((e (AllFieldHistoryStageCIncoming.stageCActive h phi ht)).val j)).val) : | |
| residentMask allocation m ε ((Work.stageC h).source, phi) side | |
| (incomingWord allocation m (AllFieldHistoryStageCIncoming.stageCActive h phi ht) w) := by | |
| intro _ | |
| have hraw := AllFieldHistoryStageCIncoming.typeWindow_of_target_side_weights | |
| allocation hm h phi ht e w side hweights (ε / 256) (div_nonneg hε (by norm_num)) | |
| have hword : | |
| (fun j => CWCompleteStatistics.twoStatistic | |
| (joinHalves (.stageC h) | |
| (w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j))) = | |
| (fun j => CWCompleteStatistics.twoStatistic | |
| ![(w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).1 (0 : Fin 1), | |
| (w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).2 (0 : Fin 1)]) := by | |
| funext j | |
| exact congrArg CWCompleteStatistics.twoStatistic (append_singletons _ _) | |
| rw [← hword] at hraw | |
| have htransport := (JointCanonicalization.typeWindow_reindex | |
| (fun a : Fin 6 => (littleLaw (cParameterParent h) (cShapeParent h) (phi.symm side) a : ℝ)) | |
| (ε / 256) | |
| (fun j => CWCompleteStatistics.twoStatistic | |
| (joinHalves (.stageC h) | |
| (w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j))) | |
| (sourcePositionsEquiv allocation m (AllFieldHistoryStageCIncoming.stageCActive h phi ht)).symm).2 hraw | |
| exact typeWindow_mono_decidable _ _ _ le_rfl _ htransport | |
| def stageCWeights (allocation : Allocation) (m : ℕ) (side : Fin 3) | |
| (e : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m)) | |
| (w : ActiveRawPairs (K := K) (tick := tick) allocation m) : Prop := | |
| ∀ (h : PartC K) (phi : Placement) | |
| (ht : FiniteSchedule.lotTick (Work.stageC h).lot (Work.stageC h).stage = tick) j, | |
| CWLeafStatistics.weight ((w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).1 (0 : Fin 1)) = | |
| (JointPopulation.shapeSide side ((e (AllFieldHistoryStageCIncoming.stageCActive h phi ht)).val j)).val ∧ | |
| CWLeafStatistics.weight ((w (AllFieldHistoryStageCIncoming.stageCActive h phi ht) j).2 (0 : Fin 1)) = | |
| activeParentShape (AllFieldHistoryStageCIncoming.stageCActive h phi ht) side - | |
| (JointPopulation.shapeSide side ((e (AllFieldHistoryStageCIncoming.stageCActive h phi ht)).val j)).val | |
| theorem received_of_rawPass (allocation : Allocation) {m : ℕ} (hm : 0 < m) | |
| {ε : ℝ} (hε : 0 ≤ ε) (side : Fin 3) | |
| (e : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m)) | |
| (w : ActiveRawPairs (K := K) (tick := tick) allocation m) | |
| (hw : rawPass allocation m ε side w) (hc : stageCWeights allocation m side e w) : | |
| received allocation m ε side w := by | |
| rintro ⟨⟨work, phi⟩, ht⟩ | |
| cases work with | |
| | stageA h => exact incoming_stageA allocation m ε side w h phi ht | |
| | stageB h => exact incoming_stageB allocation m hε side w hw h phi ht | |
| | stageC h => exact incoming_stageC allocation hm hε side e w h phi ht (hc h phi ht) | |
| end MatrixMultiplication.AllFieldHistoryIncomingMasks | |
| end | |
| end OAI | |