openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /FixedSourceChildCollision.lean
| import OAI.Computability.PerfectCompleteness.Decoding.FixedProjectionErrorLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.SourceChildMarkedLaw | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedSourceChildCollision | |
| noncomputable section | |
| open scoped Classical BigOperators | |
| open FixedParameters FixedRows RecursiveSpaces DescendantSpaces TreeSourceSpaces | |
| open UniqueGamesTheorem.Foundations.Games | |
| variable {δ : ℚ} {hδ : 0 < δ} (parameters : Parameters δ hδ) | |
| {n height v m : Nat} [NeZero m] | |
| (path : Path (branch parameters) n (height + 1)) | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch parameters height) → Slots (branch parameters) height) | |
| abbrev Calls := Option (WholeCutCalls.Index (rows parameters.plan) (repeats parameters.plan) path) | |
| abbrev Questions := SourceChildMarkedLaw.Questions | |
| (branch := branch parameters) (n := height) | |
| (t := sourceLength parameters.plan hδ) (m := m) | |
| abbrev Native (q : Questions (m := m) parameters (height := height)) := | |
| CutChildGrouping.Raw (C := Calls parameters path) | |
| (SourceChildMarkedLaw.questionSlots clauses q) (rows parameters.plan) | |
| abbrev Form (q : Questions (m := m) parameters (height := height)) := | |
| ChildBilinearCollisionTransfer.ParentForm (SourceChildMarkedLaw.questionSlots clauses q) | |
| abbrev Sample := SourceChildKernel.Sample (C := Calls parameters path) | |
| (t := sourceLength parameters.plan hδ) (rows parameters.plan) clauses designated | |
| def markedLaw : FiniteDistribution (Sample parameters path clauses designated) := | |
| SourceChildKernel.originalLaw (C := Calls parameters path) | |
| (t := sourceLength parameters.plan hδ) (rows parameters.plan) clauses designated | |
| (fun _ => ProjectionPosterior.bernoulli (projectionProbability parameters height : ℝ) | |
| (by exact_mod_cast (projectionProbability_bounds parameters height).1.le) | |
| (by exact_mod_cast (projectionProbability_bounds parameters height).2.le)) | |
| def uniformLaw : FiniteDistribution | |
| (Σ q : Questions (m := m) parameters (height := height), Native parameters path clauses q) := | |
| CompletionSoundness.sigmaLaw | |
| (PreliminarySampler.questionsLaw (branch := branch parameters) (n := height + 1) | |
| (t := sourceLength parameters.plan hδ) (m := m)) | |
| (fun q => CutChildGrouping.rawLaw (C := Calls parameters path) | |
| (SourceChildMarkedLaw.questionSlots clauses q) (rows parameters.plan)) | |
| variable (F₀ F₁ : (q : Questions (m := m) parameters (height := height)) → | |
| Native parameters path clauses q → Option (Form parameters clauses q)) | |
| def restrictedEvent (sample : Sample parameters path clauses designated) : Bool := | |
| let record := SourceChildMarkedLaw.observe (rows parameters.plan) clauses designated sample | |
| BilinearCollisionTransfer.restrictedCollision | |
| (F₀ record.1 record.2.2) (F₁ record.1 record.2.2) | |
| (ChildUnmarkedSpace.space (SourceChildMarkedLaw.questionSlots clauses record.1) record.2.1) | |
| def fullEvent (record : | |
| Σ q : Questions (m := m) parameters (height := height), Native parameters path clauses q) : Bool := | |
| BilinearCollisionTransfer.fullCollision (F₀ record.1 record.2) (F₁ record.1 record.2) | |
| private theorem expectation_add_const {A : Type*} [Fintype A] | |
| (μ : FiniteDistribution A) (f : A → ℝ) (c : ℝ) : | |
| μ.expectation (fun x => f x + c) = μ.expectation f + c := by | |
| simp only [FiniteDistribution.expectation, mul_add, Finset.sum_add_distrib, | |
| ← Finset.sum_mul, μ.normalized, one_mul] | |
| theorem restricted_probability_le (hn : n ≤ parameters.plan.depth) : | |
| (markedLaw parameters path clauses designated).probability | |
| (restrictedEvent parameters path clauses designated F₀ F₁) ≤ | |
| (uniformLaw parameters path clauses).probability | |
| (fullEvent parameters path clauses F₀ F₁) + 2 * parameters.accuracy := by | |
| have hβ : 0 ≤ (projectionProbability parameters height : ℝ) := by | |
| exact_mod_cast (projectionProbability_bounds parameters height).1.le | |
| have hβ' : (projectionProbability parameters height : ℝ) ≤ 1 := by | |
| exact_mod_cast (projectionProbability_bounds parameters height).2.le | |
| have hlaw := SourceChildMarkedLaw.original_extended_marked_law | |
| (t := sourceLength parameters.plan hδ) (rows parameters.plan) clauses designated | |
| (repeats parameters.plan) path (projectionProbability parameters height : ℝ) hβ hβ' | |
| have hprob := congrArg | |
| (fun μ : FiniteDistribution | |
| (SourceChildMarkedLaw.Observation (C := Calls parameters path) | |
| (t := sourceLength parameters.plan hδ) (rows parameters.plan) clauses) => | |
| μ.probability (fun record => BilinearCollisionTransfer.restrictedCollision | |
| (F₀ record.1 record.2.2) (F₁ record.1 record.2.2) | |
| (ChildUnmarkedSpace.space (SourceChildMarkedLaw.questionSlots clauses record.1) | |
| record.2.1))) hlaw | |
| rw [FiniteDistribution.probability_pushforward, | |
| CompletionSoundness.sigmaLaw_probability] at hprob | |
| rw [show (markedLaw parameters path clauses designated).probability | |
| (restrictedEvent parameters path clauses designated F₀ F₁) = _ from hprob] | |
| rw [uniformLaw, CompletionSoundness.sigmaLaw_probability] | |
| have hpoint := fun q : Questions (m := m) parameters (height := height) => | |
| (FixedChildCollisionTransfer.marked_restricted_lt_uniform_full parameters | |
| (FixedChildCollisionTransfer.lower_pair_calls_le parameters path hn) | |
| (SourceChildMarkedLaw.questionSlots clauses q) | |
| (SourceChildMarkedLaw.projectedSlots clauses designated q) | |
| (SourceChildMarkedLaw.questionProjection clauses designated q) | |
| (fun _ => FiniteDistribution.uniform | |
| (SourceChildMarkedLaw.Choice (branch := branch parameters) (n := height) | |
| (t := sourceLength parameters.plan hδ))) | |
| (F₀ q) (F₁ q)).le | |
| have havg := SmallBias.expectation_mono | |
| (PreliminarySampler.questionsLaw (branch := branch parameters) (n := height + 1) | |
| (t := sourceLength parameters.plan hδ) (m := m)) hpoint | |
| rw [expectation_add_const] at havg | |
| simpa only [← SourceChildMarkedLaw.replacementLaws_eq, fullEvent] using havg | |
| end | |
| end PerfectCompleteness.FixedSourceChildCollision | |
| end | |
| end OAI | |