openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Foundations /HierarchicalUsefulBackground.lean
| import OAI.Computability.PerfectCompleteness.Decoding.HierarchicalUsefulCollision | |
| import OAI.Computability.PerfectCompleteness.Foundations.DependentPredictionDifferenceLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.HierarchicalBackgroundVariation | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.HierarchicalUsefulBackground | |
| open TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| open scoped Classical | |
| noncomputable section | |
| attribute [local instance] linearMapFintype | |
| variable {branch rows : Nat → Nat} {n t : Nat} | |
| (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch n) | |
| local instance backgroundFintype : | |
| Fintype (HierarchicalAgreementMean.Background (rows := rows) slots upper) := Fintype.ofFinite _ | |
| local instance rowSpaceFintype : Fintype (NodeEmbedding.RowSpace slots upper) := Fintype.ofFinite _ | |
| variable {P : Type*} [Fintype P] | |
| (externalLaw : FiniteDistribution P) | |
| (background : P → HierarchicalAgreementMean.Background (rows := rows) slots upper) | |
| (scalarLaw : P → FiniteDistribution (NodeEmbedding.RowSpace slots upper)) | |
| (hrows : 0 < rows (Nodes.height upper)) | |
| theorem pairLaw_background : | |
| (HierarchicalUsefulCollision.pairLaw slots upper externalLaw background scalarLaw hrows).pushforward | |
| Prod.fst = externalLaw.pushforward background := by | |
| apply SigmaObservation.eq_of_probability_eq | |
| intro event | |
| rw [FiniteDistribution.probability_pushforward, HierarchicalUsefulCollision.pairLaw, | |
| FiniteDistribution.probability_pushforward, FiniteDistribution.probability_pushforward] | |
| change (CompletionSoundness.sigmaLaw externalLaw | |
| (fun p => HierarchicalUsefulCollision.bucketLaw slots upper (scalarLaw p) hrows)).probability | |
| (fun sample => event (background sample.1)) = | |
| externalLaw.probability (fun p => event (background p)) | |
| exact CompletionSoundness.sigmaLaw_probability_first externalLaw | |
| (fun p => HierarchicalUsefulCollision.bucketLaw slots upper (scalarLaw p) hrows) | |
| (fun p => event (background p)) | |
| theorem original_background_variation | |
| (referenceBackground : FiniteDistribution | |
| (HierarchicalAgreementMean.Background (rows := rows) slots upper)) : | |
| (HierarchicalUsefulCollision.pairLaw slots upper externalLaw background scalarLaw hrows).totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots upper | |
| (externalLaw.pushforward background) hrows) ≤ | |
| 2 * (HierarchicalUsefulCollision.pairLaw slots upper externalLaw background scalarLaw hrows).totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots upper referenceBackground hrows) := by | |
| have h := HierarchicalBackgroundVariation.original_background_variation slots upper | |
| (HierarchicalUsefulCollision.pairLaw slots upper externalLaw background scalarLaw hrows) | |
| referenceBackground hrows | |
| rw [pairLaw_background] at h | |
| exact h | |
| theorem original_background_le_of_bound | |
| (referenceBackground : FiniteDistribution | |
| (HierarchicalAgreementMean.Background (rows := rows) slots upper)) {error : ℝ} | |
| (herror : (HierarchicalUsefulCollision.pairLaw slots upper externalLaw background scalarLaw hrows).totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots upper referenceBackground hrows) ≤ error) : | |
| (HierarchicalUsefulCollision.pairLaw slots upper externalLaw background scalarLaw hrows).totalVariation | |
| (HierarchicalAgreementMean.referenceLaw slots upper | |
| (externalLaw.pushforward background) hrows) ≤ 2 * error := by | |
| have h := HierarchicalBackgroundVariation.original_background_le_of_bound slots upper | |
| (HierarchicalUsefulCollision.pairLaw slots upper externalLaw background scalarLaw hrows) | |
| referenceBackground hrows herror | |
| rw [pairLaw_background] at h | |
| exact h | |
| end | |
| end PerfectCompleteness.HierarchicalUsefulBackground | |
| end | |
| end OAI | |