openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /StoppedBucketUsefulProbability.lean
| import OAI.Computability.PerfectCompleteness.Construction.HiddenBucketBias | |
| import OAI.Computability.PerfectCompleteness.Reduction.PreliminaryAveraging | |
| import OAI.Computability.PerfectCompleteness.Sampling.StoppedUsefulPairLaw | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.StoppedBucketUsefulProbability | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open OriginalWholeCutTape WholeArrayInteriorExterior | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| attribute [local instance] linearMapFintype | |
| private theorem probability_product_fst {A B : Type*} [Fintype A] [Fintype B] | |
| (μ : FiniteDistribution A) (ν : FiniteDistribution B) (event : A → Bool) : | |
| (μ.product ν).probability (fun z => event z.1) = μ.probability event := by | |
| have h := FiniteDistribution.probability_pushforward (μ.product ν) Prod.fst event | |
| rw [HiddenBucketBias.product_fst] at h | |
| exact h.symm | |
| variable {branch : Nat → Nat} {N j i t : Nat} | |
| (rows repeats : Nat → Nat) (p : Path branch N (j + 1)) | |
| (chosen : Fin (branch j)) (q : Path branch j i) | |
| (native right : Slots branch N → Fin t → MixedSupport.Slot) | |
| (projection : ∀ s a, MixedSupport.Projection (native s a) (right s a)) | |
| local instance rowSpaceFintype : Fintype (NodeEmbedding.RowSpace native (upperNode p)) := | |
| Fintype.ofFinite _ | |
| local instance backgroundFintype : | |
| Fintype (HierarchicalMatrixTable.Background (rows := rows) native (upperNode p)) := | |
| Fintype.ofFinite _ | |
| variable (hrows : 0 < rows (j + 1)) (lowerLevel : Nat) {Ω : Type*} [Fintype Ω] | |
| (original : FiniteDistribution Ω) (arrays : Ω → Arrays native rows) | |
| (lowerEvent : Ω → Bool) (κ : ℝ) | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch N t)) | |
| theorem usefulEvent_observe | |
| (sample : StoppedUsefulPairLaw.Sample rows repeats p chosen q right) : | |
| HierarchicalUsefulCollision.usefulEvent native (upperNode p) lowerLevel | |
| original arrays lowerEvent κ σ | |
| (StoppedUsefulPairLaw.background rows repeats p chosen q native right projection) | |
| (StoppedUsefulPairLaw.observe rows repeats p chosen q native right projection sample) = | |
| HierarchicalUsefulCollision.usefulAccepted native (upperNode p) lowerLevel | |
| original arrays lowerEvent κ σ | |
| (StoppedUsefulPairLaw.background rows repeats p chosen q native right projection | |
| (StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection sample.2.1).1) | |
| (CutNodeRows.directionEquiv rows p sample.1) | |
| (CutNodeRows.bucketTapeEquiv rows p native | |
| (StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection sample.2.1).2) := | |
| rfl | |
| theorem useful_probability_eq : | |
| (HierarchicalUsefulCollision.law native (upperNode p) | |
| (StoppedUsefulPairLaw.externalLaw rows repeats p chosen q right) | |
| (StoppedUsefulPairLaw.scalarLaw rows repeats p chosen q native right projection) | |
| (by simpa only [upperNode_height] using hrows)).probability | |
| (HierarchicalUsefulCollision.usefulEvent native (upperNode p) lowerLevel | |
| original arrays lowerEvent κ σ | |
| (StoppedUsefulPairLaw.background rows repeats p chosen q native right projection)) = | |
| (StoppedUsefulPairLaw.directionLaw rows j hrows).expectation (fun a => | |
| (WholeArraySampler.tapeLaw rows repeats (p.append (.step chosen q)) right).probability | |
| (fun tape => HierarchicalUsefulCollision.usefulAccepted native (upperNode p) lowerLevel | |
| original arrays lowerEvent κ σ | |
| (StoppedUsefulPairLaw.background rows repeats p chosen q native right projection | |
| (StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection tape).1) | |
| (CutNodeRows.directionEquiv rows p a) | |
| (CutNodeRows.bucketTapeEquiv rows p native | |
| (StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection tape).2))) := by | |
| rw [← StoppedUsefulPairLaw.observe_law rows repeats p chosen q native right projection hrows, | |
| FiniteDistribution.probability_pushforward] | |
| simp_rw [usefulEvent_observe] | |
| rw [StoppedUsefulPairLaw.actualLaw, PreliminaryAveraging.probability_product] | |
| apply FiniteDistribution.expectation_congr | |
| intro a | |
| exact probability_product_fst | |
| (WholeArraySampler.tapeLaw rows repeats (p.append (.step chosen q)) right) | |
| (RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain (cutSlots p right))) | |
| (fun tape => HierarchicalUsefulCollision.usefulAccepted native (upperNode p) lowerLevel | |
| original arrays lowerEvent κ σ | |
| (StoppedUsefulPairLaw.background rows repeats p chosen q native right projection | |
| (StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection tape).1) | |
| (CutNodeRows.directionEquiv rows p a) | |
| (CutNodeRows.bucketTapeEquiv rows p native | |
| (StoppedUsefulPairLaw.readout rows repeats p chosen q native right projection tape).2)) | |
| end | |
| end PerfectCompleteness.StoppedBucketUsefulProbability | |
| end | |
| end OAI | |