openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /HierarchicalUsefulCollision.lean
| import OAI.Computability.PerfectCompleteness.Decoding.BucketUsefulCollision | |
| import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalAgreementMean | |
| import OAI.Computability.PerfectCompleteness.Reduction.CompletionSoundness | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.HierarchicalUsefulCollision | |
| noncomputable section | |
| open scoped BigOperators Classical | |
| open TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| attribute [local instance] linearMapFintype | |
| private theorem probability_product {A B : Type*} [Fintype A] [Fintype B] | |
| (μ : FiniteDistribution A) (ν : FiniteDistribution B) (event : A × B → Bool) : | |
| (μ.product ν).probability event = | |
| μ.expectation (fun a => ν.probability (fun b => event (a, b))) := by | |
| simp only [FiniteDistribution.probability, FiniteDistribution.expectation, | |
| FiniteDistribution.product, Fintype.sum_prod_type, Finset.mul_sum, mul_ite, mul_zero] | |
| private theorem probability_product_first {A B : Type*} [Fintype A] [Fintype B] | |
| (μ : FiniteDistribution A) (ν : FiniteDistribution B) (event : A → Bool) : | |
| (μ.product ν).probability (fun x => event x.1) = μ.probability event := by | |
| unfold FiniteDistribution.probability | |
| rw [Fintype.sum_prod_type] | |
| apply Finset.sum_congr rfl | |
| intro a _ | |
| cases he : event a <;> | |
| simp [FiniteDistribution.product, he, ← Finset.mul_sum, ν.normalized] | |
| variable {branch rows : Nat → Nat} {n t : Nat} | |
| (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch n) | |
| abbrev Background := HierarchicalAgreementMean.Background (rows := rows) slots upper | |
| abbrev Matrix := HierarchicalAgreementMean.Matrix (rows := rows) slots upper | |
| abbrev Direction := BucketSampler.Direction (rows (Nodes.height upper)) | |
| abbrev Tape := BucketSampler.Tape (rows (Nodes.height upper)) (NodeEmbedding.RowSpace slots upper) | |
| local instance backgroundFintype : Fintype (Background (rows := rows) slots upper) := | |
| Fintype.ofFinite _ | |
| local instance rowSpaceFintype : Fintype (NodeEmbedding.RowSpace slots upper) := | |
| Fintype.ofFinite _ | |
| local instance valueFintype : | |
| Fintype (Block rows upper × HierarchicalMatrixTable.SideOutput (rows := rows) upper) := | |
| Fintype.ofFinite _ | |
| def directionLaw (hrows : 0 < rows (Nodes.height upper)) : | |
| FiniteDistribution (Direction (rows := rows) upper) := by | |
| letI : Nonempty (Direction (rows := rows) upper) := | |
| ⟨BucketUniform.coordinateDirection ⟨0, hrows⟩⟩ | |
| exact FiniteDistribution.uniform _ | |
| def bucketLaw (ν : FiniteDistribution (NodeEmbedding.RowSpace slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) : | |
| FiniteDistribution (HierarchicalAgreementMean.BucketSample (rows := rows) slots upper) := | |
| (directionLaw upper hrows).product | |
| ((BucketSampler.tapeLaw (rows (Nodes.height upper)) ν).product ν) | |
| variable {P : Type*} [Fintype P] | |
| abbrev Sample := (_ : P) × HierarchicalAgreementMean.BucketSample (rows := rows) slots upper | |
| def law (externalLaw : FiniteDistribution P) | |
| (scalarLaw : P → FiniteDistribution (NodeEmbedding.RowSpace slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) : | |
| FiniteDistribution (Sample (P := P) (rows := rows) slots upper) := | |
| CompletionSoundness.sigmaLaw externalLaw (fun p => bucketLaw slots upper (scalarLaw p) hrows) | |
| def pairRecord (background : P → Background (rows := rows) slots upper) | |
| (sample : Sample (P := P) (rows := rows) slots upper) : | |
| HierarchicalAgreementMean.PairRecord (rows := rows) slots upper := | |
| HierarchicalAgreementMean.pairRecord slots upper (background sample.1, sample.2) | |
| def pairLaw (externalLaw : FiniteDistribution P) | |
| (background : P → Background (rows := rows) slots upper) | |
| (scalarLaw : P → FiniteDistribution (NodeEmbedding.RowSpace slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) : | |
| FiniteDistribution (HierarchicalAgreementMean.PairRecord (rows := rows) slots upper) := | |
| (law slots upper externalLaw scalarLaw hrows).pushforward (pairRecord slots upper background) | |
| variable (lowerLevel : Nat) {Ω : Type*} [Fintype Ω] | |
| (original : FiniteDistribution Ω) (arrays : Ω → Arrays slots rows) | |
| (lower : Ω → Bool) (κ : ℝ) | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t)) | |
| def usefulAccepted (background : Background (rows := rows) slots upper) | |
| (a : Direction (rows := rows) upper) (tape : Tape (rows := rows) slots upper) : Bool := | |
| decide (CanonicalMatrixTable.Accepts (TreeCanonical.numberedSlots slots) | |
| (NodeEmbedding.RowSpace slots upper) (HierarchicalMatrixTable.other slots upper background) | |
| σ a.val (BucketMatrixResampling.assembledMatrix (NodeEmbedding.RowSpace slots upper) | |
| (rows (Nodes.height upper)) tape)) && | |
| (HierarchicalUsefulness.table slots upper lowerLevel original arrays lower κ σ background | |
| (MatrixRowQuotient.projectMatrix | |
| (HierarchicalFrozenTables.knownRows slots upper lowerLevel background) | |
| (BucketMatrixResampling.assembledMatrix (NodeEmbedding.RowSpace slots upper) | |
| (rows (Nodes.height upper)) tape))).isSome | |
| def usefulEvent (background : P → Background (rows := rows) slots upper) | |
| (sample : Sample (P := P) (rows := rows) slots upper) : Bool := | |
| usefulAccepted slots upper lowerLevel original arrays lower κ σ | |
| (background sample.1) sample.2.1 sample.2.2.1 | |
| theorem direction_collision_lower_bound | |
| (background : Background (rows := rows) slots upper) | |
| (ν : FiniteDistribution (NodeEmbedding.RowSpace slots upper)) | |
| (a : Direction (rows := rows) upper) : | |
| (BucketSampler.tapeLaw (rows (Nodes.height upper)) ν).probability | |
| (usefulAccepted slots upper lowerLevel original arrays lower κ σ background a) ^ 2 / 2 ≤ | |
| ((BucketSampler.tapeLaw (rows (Nodes.height upper)) ν).product ν).probability | |
| (fun z => HierarchicalAgreementMean.collision slots upper lowerLevel | |
| original arrays lower κ σ | |
| (HierarchicalAgreementMean.pairRecord slots upper (background, (a, z)))) := by | |
| exact BucketUsefulCollision.collision_probability_lower_bound | |
| (TreeCanonical.numberedSlots slots) (NodeEmbedding.RowSpace slots upper) | |
| (HierarchicalMatrixTable.other slots upper background) σ | |
| (HierarchicalFrozenTables.knownRows slots upper lowerLevel background) | |
| (HierarchicalFrozenTables.sectionMap slots upper lowerLevel background) | |
| (HierarchicalFrozenTables.sectionMap_spec slots upper lowerLevel background) | |
| (HierarchicalFrozenTables.determined slots upper lowerLevel background) ν | |
| (fun X => HierarchicalUsefulness.mark slots upper lowerLevel original arrays lower κ | |
| ⟨background, X⟩ = true) a | |
| theorem useful_probability_eq | |
| (externalLaw : FiniteDistribution P) | |
| (background : P → Background (rows := rows) slots upper) | |
| (scalarLaw : P → FiniteDistribution (NodeEmbedding.RowSpace slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) : | |
| (law slots upper externalLaw scalarLaw hrows).probability | |
| (usefulEvent slots upper lowerLevel original arrays lower κ σ background) = | |
| externalLaw.expectation (fun p => (directionLaw upper hrows).expectation (fun a => | |
| (BucketSampler.tapeLaw (rows (Nodes.height upper)) (scalarLaw p)).probability | |
| (usefulAccepted slots upper lowerLevel original arrays lower κ σ (background p) a))) := by | |
| rw [law, CompletionSoundness.sigmaLaw_probability] | |
| apply FiniteDistribution.expectation_congr | |
| intro p | |
| rw [bucketLaw, probability_product] | |
| apply FiniteDistribution.expectation_congr | |
| intro a | |
| exact probability_product_first | |
| (BucketSampler.tapeLaw (rows (Nodes.height upper)) (scalarLaw p)) (scalarLaw p) | |
| (usefulAccepted slots upper lowerLevel original arrays lower κ σ (background p) a) | |
| theorem collision_probability_eq | |
| (externalLaw : FiniteDistribution P) | |
| (background : P → Background (rows := rows) slots upper) | |
| (scalarLaw : P → FiniteDistribution (NodeEmbedding.RowSpace slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) : | |
| (pairLaw slots upper externalLaw background scalarLaw hrows).probability | |
| (HierarchicalAgreementMean.collision slots upper lowerLevel original arrays lower κ σ) = | |
| externalLaw.expectation (fun p => (directionLaw upper hrows).expectation (fun a => | |
| ((BucketSampler.tapeLaw (rows (Nodes.height upper)) (scalarLaw p)).product | |
| (scalarLaw p)).probability (fun z => | |
| HierarchicalAgreementMean.collision slots upper lowerLevel original arrays lower κ σ | |
| (HierarchicalAgreementMean.pairRecord slots upper (background p, (a, z)))))) := by | |
| rw [pairLaw, FiniteDistribution.probability_pushforward, law, | |
| CompletionSoundness.sigmaLaw_probability] | |
| apply FiniteDistribution.expectation_congr | |
| intro p | |
| exact probability_product (directionLaw upper hrows) | |
| ((BucketSampler.tapeLaw (rows (Nodes.height upper)) (scalarLaw p)).product (scalarLaw p)) | |
| (fun z => HierarchicalAgreementMean.collision slots upper lowerLevel original arrays lower κ σ | |
| (HierarchicalAgreementMean.pairRecord slots upper (background p, z))) | |
| theorem collision_probability_lower_bound | |
| (externalLaw : FiniteDistribution P) | |
| (background : P → Background (rows := rows) slots upper) | |
| (scalarLaw : P → FiniteDistribution (NodeEmbedding.RowSpace slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) : | |
| (law slots upper externalLaw scalarLaw hrows).probability | |
| (usefulEvent slots upper lowerLevel original arrays lower κ σ background) ^ 2 / 2 ≤ | |
| (pairLaw slots upper externalLaw background scalarLaw hrows).probability | |
| (HierarchicalAgreementMean.collision slots upper lowerLevel original arrays lower κ σ) := by | |
| rw [useful_probability_eq, collision_probability_eq] | |
| apply CollisionAveraging.average_square_bound | |
| intro p | |
| apply CollisionAveraging.average_square_bound | |
| intro a | |
| exact direction_collision_lower_bound slots upper lowerLevel original arrays lower κ σ | |
| (background p) (scalarLaw p) a | |
| theorem collision_probability_ge_eighth | |
| (externalLaw : FiniteDistribution P) | |
| (background : P → Background (rows := rows) slots upper) | |
| (scalarLaw : P → FiniteDistribution (NodeEmbedding.RowSpace slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) (c : ℝ) (hc : 0 ≤ c) | |
| (hmass : c / 2 ≤ (law slots upper externalLaw scalarLaw hrows).probability | |
| (usefulEvent slots upper lowerLevel original arrays lower κ σ background)) : | |
| c ^ 2 / 8 ≤ | |
| (pairLaw slots upper externalLaw background scalarLaw hrows).probability | |
| (HierarchicalAgreementMean.collision slots upper lowerLevel original arrays lower κ σ) := by | |
| have h := collision_probability_lower_bound slots upper lowerLevel original arrays lower κ σ | |
| externalLaw background scalarLaw hrows | |
| have hprod : 0 ≤ | |
| ((law slots upper externalLaw scalarLaw hrows).probability | |
| (usefulEvent slots upper lowerLevel original arrays lower κ σ background) - c / 2) * | |
| ((law slots upper externalLaw scalarLaw hrows).probability | |
| (usefulEvent slots upper lowerLevel original arrays lower κ σ background) + c / 2) := | |
| mul_nonneg (sub_nonneg.mpr hmass) (by linarith) | |
| nlinarith | |
| end | |
| end PerfectCompleteness.HierarchicalUsefulCollision | |
| end | |
| end OAI | |