openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /PreliminaryAveraging.lean
| import OAI.Computability.PerfectCompleteness.Sampling.PreliminarySampler | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.PreliminaryAveraging | |
| open PreliminarySampler TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| theorem probability_product {X Y : Type*} [Fintype X] [Fintype Y] | |
| (μ : FiniteDistribution X) (ν : FiniteDistribution Y) (event : X × Y → Bool) : | |
| (μ.product ν).probability event = | |
| μ.expectation (fun x => ν.probability (fun y => event (x, y))) := by | |
| simp only [FiniteDistribution.probability, FiniteDistribution.expectation, | |
| FiniteDistribution.product, Fintype.sum_prod_type, Finset.mul_sum, mul_ite, mul_zero] | |
| variable {v m n t : Nat} {branch rows repeats : Nat → Nat} | |
| def directionLaw (leaf : RecursiveSpaces.Slots branch n) (level : Fin n) | |
| (hrows : ∀ k, 0 < rows (k + 1)) : | |
| FiniteDistribution (BucketSampler.Direction | |
| (rows (Nodes.height (GeometricPath.nodeAtLevel leaf level)))) := by | |
| letI : Nonempty (BucketSampler.Direction | |
| (rows (Nodes.height (GeometricPath.nodeAtLevel leaf level)))) := | |
| directionNonempty _ (by rw [GeometricPath.nodeAtLevel_height]; exact hrows level.val) | |
| exact FiniteDistribution.uniform _ | |
| theorem choice_probability (leaf : RecursiveSpaces.Slots branch n) (hn : 0 < n) | |
| (hrows : ∀ k, 0 < rows (k + 1)) (event : Choice rows leaf → Bool) : | |
| (choiceLaw leaf hn hrows).probability event = | |
| (∑ level : Fin n, (directionLaw leaf level hrows).probability | |
| (fun direction => event ⟨level, direction⟩)) / (n : ℝ) := by | |
| let : Nonempty (Fin n) := ⟨⟨0, hn⟩⟩ | |
| change (CompletionSoundness.sigmaLaw (FiniteDistribution.uniform (Fin n)) | |
| (fun level => directionLaw leaf level hrows)).probability event = _ | |
| rw [CompletionSoundness.sigmaLaw_probability, FiniteDistribution.expectation_uniform] | |
| simp only [Fintype.card_fin] | |
| theorem array_observation_probability [NeZero m] | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (rows repeats : Nat → Nat) (hn : 0 < n) | |
| (hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) | |
| (observe : ∀ b : Base branch n t m, | |
| Arrays (sourceSlots clauses (endpoints b.1)) rows → Choice rows b.2 → Bool) : | |
| (law clauses rows repeats hn hbranch hrows).probability | |
| (fun e => observe e.1 | |
| (WholeArraySampler.evaluate rows repeats (GeometricPath.leafPath e.1.2) | |
| (sourceSlots clauses (endpoints e.1.1)) e.2.1) e.2.2) = | |
| (baseLaw hbranch).expectation (fun b => | |
| (WholeArraySampler.law rows repeats (GeometricPath.leafPath b.2) | |
| (sourceSlots clauses (endpoints b.1))).expectation | |
| (fun arrays => (choiceLaw b.2 hn hrows).probability (observe b arrays))) := by | |
| rw [law_probability] | |
| apply FiniteDistribution.expectation_congr | |
| intro b | |
| rw [conditionalLaw, probability_product] | |
| rw [WholeArraySampler.law, FiniteDistribution.expectation_pushforward] | |
| theorem candidate_average [NeZero m] | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (rows repeats : Nat → Nat) (hn : 0 < n) | |
| (hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) | |
| (observe : ∀ b : Base branch n t m, | |
| Arrays (sourceSlots clauses (endpoints b.1)) rows → Choice rows b.2 → Bool) : | |
| (law clauses rows repeats hn hbranch hrows).probability | |
| (fun e => observe e.1 | |
| (WholeArraySampler.evaluate rows repeats (GeometricPath.leafPath e.1.2) | |
| (sourceSlots clauses (endpoints e.1.1)) e.2.1) e.2.2) = | |
| (baseLaw hbranch).expectation (fun b => | |
| (WholeArraySampler.law rows repeats (GeometricPath.leafPath b.2) | |
| (sourceSlots clauses (endpoints b.1))).expectation (fun arrays => | |
| (∑ level : Fin n, (directionLaw b.2 level hrows).probability | |
| (fun direction => observe b arrays ⟨level, direction⟩)) / (n : ℝ))) := by | |
| rw [array_observation_probability clauses rows repeats hn hbranch hrows observe] | |
| apply FiniteDistribution.expectation_congr | |
| intro b | |
| apply FiniteDistribution.expectation_congr | |
| intro arrays | |
| exact choice_probability b.2 hn hrows (observe b arrays) | |
| end | |
| end PerfectCompleteness.PreliminaryAveraging | |
| end | |
| end OAI | |