openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /OriginalCutPairLaw.lean
| import OAI.Computability.PerfectCompleteness.Construction.CutBucketTransportLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.OriginalCutBucketLaw | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.OriginalCutPairLaw | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open OriginalWholeCutTape WholeArrayInteriorExterior | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| open scoped Classical | |
| noncomputable section | |
| attribute [local instance] linearMapFintype | |
| variable {branch : Nat → Nat} {n m k t : Nat} | |
| local instance backgroundFintype (rows : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) : | |
| Fintype (HierarchicalMatrixTable.Background (rows := rows) slots (upperNode p)) := | |
| Fintype.ofFinite _ | |
| local instance rowSpaceFintype (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) : | |
| Fintype (NodeEmbedding.RowSpace slots (upperNode p)) := Fintype.ofFinite _ | |
| theorem matrix_reconstruct (rows repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (record : OriginalWholeCut.Record rows repeats p slots) : | |
| NodeEmbedding.matrix (OriginalWholeCut.reconstructRecord rows repeats p slots record) | |
| (upperNode p) = | |
| BucketMatrixResampling.assembledMatrix (NodeEmbedding.RowSpace slots (upperNode p)) | |
| (rows (Nodes.height (upperNode p))) | |
| (CutNodeRows.bucketTapeEquiv rows p slots | |
| (OriginalCutBucketReplacement.ownValues rows repeats p slots record.2.1)) := by | |
| rw [← CutNodeRows.matrixOfRows_selectedRows] | |
| change CutNodeRows.matrixOfRows rows p slots | |
| (SelectedArrayReplacement.selectedRows rows p slots | |
| (OriginalWholeCut.reconstruct rows repeats p slots record.1 record.2.1 record.2.2)) = _ | |
| rw [OriginalCutBucketReplacement.selectedRows_reconstruct] | |
| exact CutNodeRows.matrixOfRows_evaluate rows p slots _ | |
| def transportPair (rows : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (sample : OriginalCutBucketLaw.NativeSample rows p slots) : | |
| OriginalCutCollision.PairRecord rows p slots := | |
| HierarchicalAgreementMean.pairRecord slots (upperNode p) | |
| (sample.1, CutNodeRows.bucketSampleEquiv rows p slots sample.2) | |
| private theorem transportPair_law (rows : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (μ : FiniteDistribution (HierarchicalMatrixTable.Background (rows := rows) slots (upperNode p))) | |
| (ν : FiniteDistribution (BucketSampler.Direction (rows (m + 1)) × | |
| (BucketSampler.Tape (rows (m + 1)) (H (cutSlots p slots)) × H (cutSlots p slots)))) : | |
| (μ.product ν).pushforward (transportPair rows p slots) = | |
| ((μ.product ν).pushforward | |
| (fun z : OriginalCutBucketLaw.NativeSample rows p slots => | |
| (z.1, CutNodeRows.bucketSampleEquiv rows p slots z.2))).pushforward | |
| (HierarchicalAgreementMean.pairRecord slots (upperNode p)) := by | |
| exact (FiniteDistribution.pushforward_comp (μ.product ν) | |
| (fun z : OriginalCutBucketLaw.NativeSample rows p slots => | |
| (z.1, CutNodeRows.bucketSampleEquiv rows p slots z.2)) | |
| (HierarchicalAgreementMean.pairRecord slots (upperNode p))).symm | |
| theorem pairRecord_eq (rows repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (sample : BucketSampler.Direction (rows (m + 1)) × | |
| OriginalExtraCutComparison.Record rows repeats p slots) : | |
| OriginalCutCollision.pairRecord rows repeats p slots sample = | |
| transportPair rows p slots (OriginalCutBucketLaw.read rows repeats p slots sample) := by | |
| apply Prod.ext | |
| · rfl | |
| · apply Prod.ext | |
| · change NodeEmbedding.matrix | |
| (OriginalWholeCut.reconstructRecord rows repeats p slots | |
| (OriginalCutCollision.oldRecord rows repeats p slots sample.2)) (upperNode p) = _ | |
| rw [matrix_reconstruct] | |
| rfl | |
| · change NodeEmbedding.matrix | |
| (OriginalWholeCut.reconstructRecord rows repeats p slots | |
| (OriginalCutBucketReplacement.replaceRecord rows repeats p slots | |
| (OriginalCutCollision.oldRecord rows repeats p slots sample.2) sample.1 sample.2.2)) | |
| (upperNode p) = _ | |
| rw [matrix_reconstruct] | |
| change BucketMatrixResampling.assembledMatrix (NodeEmbedding.RowSpace slots (upperNode p)) | |
| (rows (Nodes.height (upperNode p))) | |
| (CutNodeRows.bucketTapeEquiv rows p slots | |
| (OriginalCutBucketReplacement.ownValues rows repeats p slots | |
| (OriginalCutBucketReplacement.replaceValues rows repeats p slots | |
| (OriginalCutCollision.oldRecord rows repeats p slots sample.2).2.1 | |
| sample.1 sample.2.2))) = _ | |
| rw [OriginalCutBucketReplacement.ownValues_replaceValues, CutNodeRows.bucketTapeEquiv_update] | |
| rfl | |
| def directionsLaw (rows : Nat → Nat) (m : Nat) (hrows : 0 < rows (m + 1)) : | |
| FiniteDistribution (BucketSampler.Direction (rows (m + 1))) := by | |
| letI : Nonempty (BucketSampler.Direction (rows (m + 1))) := | |
| ⟨BucketUniform.coordinateDirection ⟨0, hrows⟩⟩ | |
| exact FiniteDistribution.uniform _ | |
| theorem reference_pair_law (rows repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (hrows : 0 < rows (m + 1)) : | |
| ((directionsLaw rows m hrows).product | |
| (OriginalExtraCutComparison.referenceLaw rows repeats p slots)).pushforward | |
| (OriginalCutCollision.pairRecord rows repeats p slots) = | |
| HierarchicalAgreementMean.referenceLaw slots (upperNode p) | |
| (OriginalOwnBucketSplit.backgroundLaw rows repeats p slots) | |
| (by simpa only [upperNode_height] using hrows) := by | |
| let μ := OriginalOwnBucketSplit.backgroundLaw rows repeats p slots | |
| have hread := OriginalCutBucketLaw.reference_read_law rows repeats p slots | |
| (directionsLaw rows m hrows) | |
| have hnative : | |
| ((directionsLaw rows m hrows).product | |
| (OriginalExtraCutComparison.referenceLaw rows repeats p slots)).pushforward | |
| (OriginalCutBucketLaw.read rows repeats p slots) = | |
| μ.product (CutBucketTransport.nativeLaw rows p slots hrows) := by | |
| simpa only [μ, CutBucketTransport.nativeLaw, directionsLaw, BucketSampler.tapeLaw, UniformLinearImage.law_uniform] using hread | |
| have htransport := FiniteDistribution.product_pushforward μ | |
| (CutBucketTransport.nativeLaw rows p slots hrows) id | |
| (CutNodeRows.bucketSampleEquiv rows p slots) | |
| simp only [id_eq, FiniteDistribution.pushforward_id] at htransport | |
| calc | |
| _ = (((directionsLaw rows m hrows).product | |
| (OriginalExtraCutComparison.referenceLaw rows repeats p slots)).pushforward | |
| (OriginalCutBucketLaw.read rows repeats p slots)).pushforward | |
| (transportPair rows p slots) := by | |
| rw [FiniteDistribution.pushforward_comp] | |
| congr 1 | |
| funext sample | |
| exact pairRecord_eq rows repeats p slots sample | |
| _ = (μ.product (CutBucketTransport.nativeLaw rows p slots hrows)).pushforward | |
| (transportPair rows p slots) := by rw [hnative] | |
| _ = ((μ.product (CutBucketTransport.nativeLaw rows p slots hrows)).pushforward | |
| (fun z => (z.1, CutNodeRows.bucketSampleEquiv rows p slots z.2))).pushforward | |
| (HierarchicalAgreementMean.pairRecord slots (upperNode p)) := | |
| transportPair_law rows p slots μ (CutBucketTransport.nativeLaw rows p slots hrows) | |
| _ = _ := by | |
| rw [htransport, CutBucketTransport.nativeLaw_pushforward] | |
| rfl | |
| theorem actual_pair_totalVariation_le {K : Type*} [Fintype K] | |
| (rows repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (ν : FiniteDistribution K) (q : K → Path branch m k) (hbranch : 0 < branch m) | |
| (hrows : 0 < rows (m + 1)) : | |
| (((directionsLaw rows m hrows).product | |
| (OriginalExtraCutComparison.actualLaw rows repeats p slots ν q hbranch)).pushforward | |
| (OriginalCutCollision.pairRecord rows repeats p slots)).totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots (upperNode p) | |
| (OriginalOwnBucketSplit.backgroundLaw rows repeats p slots) | |
| (by simpa only [upperNode_height] using hrows)) ≤ | |
| Real.sqrt ((ChildBlockCardinality.bound branch m t | |
| (OriginalCutCalls.count rows repeats n (m + 1) + 1) rows : ℝ) ^ 2 / branch m) / 2 := by | |
| rw [← reference_pair_law rows repeats p slots hrows] | |
| exact OriginalCutCollision.pair_totalVariation_le rows repeats p slots ν q hbranch | |
| (directionsLaw rows m hrows) | |
| end | |
| end PerfectCompleteness.OriginalCutPairLaw | |
| end | |
| end OAI | |