openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Repetition /AmplifiedCleanRateLemmas.lean
| import OAI.Computability.PerfectCompleteness.Reduction.OddListGame | |
| import OAI.Computability.PerfectCompleteness.Repetition.CutChildCleanLawLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.ProjectedCleanRate | |
| open scoped BigOperators Classical | |
| open UniqueGamesTheorem.Foundations.Games | |
| open RecursiveSpaces ProjectedCardinality CleanTreeFactors UniformCleanSources | |
| noncomputable section | |
| section | |
| variable {v m s L cube : Nat} [NeZero m] [NeZero s] | |
| (clauses : Fin m → SourceClause.NormalizedClause v) (γ : ℚ) | |
| (hγ : 0 < γ) (hγ1 : γ ≤ 1) | |
| {R D : Fin (cube ^ 3) → Type*} | |
| [∀ i, Fintype (R i)] [∀ i, DecidableEq (R i)] | |
| [∀ i, Fintype (D i)] [∀ i, DecidableEq (D i)] | |
| (branch : Nat → Nat) (height calls : Nat) (rows : Nat → Nat) | |
| variable (ids : (i : Fin (cube ^ 3)) → | |
| SourceOddLists.Occurrences m (SourceOddLists.repetitionLength γ s hγ hγ1) × R i → Slots branch height → Fin (SourceOddLists.repetitionLength γ s hγ hγ1) → Nat) | |
| local notation "Ωsource" => Factors branch height (SourceOddLists.repetitionLength γ s hγ hγ1) calls rows ids | |
| local notation "zsource" => zeros branch height (SourceOddLists.repetitionLength γ s hγ hγ1) calls rows ids | |
| omit [∀ index, DecidableEq (D index)] in | |
| theorem amplified_source_bound (hcube : 0 < cube) | |
| (gap : SourceAmplification.ClauseGap clauses γ) | |
| (hL : 0 < L) (alphabet : SourceOddLists.alphabetLog s (SourceOddLists.repetitionLength γ s hγ hγ1) ≤ L) | |
| (ν : (i : Fin (cube ^ 3)) → FiniteDistribution (R i)) | |
| (nonprojected : (i : Fin (cube ^ 3)) → | |
| SourceOddLists.Occurrences m (SourceOddLists.repetitionLength γ s hγ hγ1) × R i → FiniteDistribution (D i)) | |
| (event : UniformCleanSoundness.Sample (D := D) Ωsource → Bool) | |
| (strategies : (record : UniformCleanSoundness.FullRecord (D := D) Ωsource) → | |
| Strategy (CleanIndices Ωsource record → Fin (SourceOddLists.repetitionLength γ s hγ hγ1) → Fin m) | |
| (CleanIndices Ωsource record → Fin (SourceOddLists.repetitionLength γ s hγ hγ1) → Fin v) | |
| (CleanIndices Ωsource record → OddLists.OddList s (SourceOddLists.LeftLabels (SourceOddLists.repetitionLength γ s hγ hγ1))) | |
| (CleanIndices Ωsource record → OddLists.OddList s (SourceOddLists.RightLabels (SourceOddLists.repetitionLength γ s hγ hγ1)))) | |
| (inclusion : ∀ record x, recordEvent (D := D) Ωsource zsource record x = true → | |
| event x = true → | |
| (IndexedRepetition.game (SourceOddLists.game s clauses (SourceOddLists.repetitionLength γ s hγ hγ1)) | |
| (CleanIndices Ωsource record)).wins (strategies record) | |
| (fun i => (x i.1).1) = true) : | |
| (childrenLaw Ωsource (fun _ => SourceOddLists.tupleLaw m (SourceOddLists.repetitionLength γ s hγ hγ1)) ν | |
| (fun _ => projectionFlag cube hcube) nonprojected).probability event ≤ | |
| (1 - coefficient L branch height (SourceOddLists.repetitionLength γ s hγ hγ1) calls rows / (cube : ℝ) ^ 2) ^ (cube ^ 3) := by | |
| exact bound_from_repetition branch height (SourceOddLists.repetitionLength γ s hγ hγ1) calls rows ids hcube | |
| (SourceOddLists.game s clauses (SourceOddLists.repetitionLength γ s hγ hγ1)) | |
| (SourceOddLists.repetition_halfRate clauses γ hγ hγ1 gap hL alphabet) | |
| ν nonprojected event strategies inclusion | |
| end | |
| end | |
| end PerfectCompleteness.ProjectedCleanRate | |
| end | |
| section | |
| namespace PerfectCompleteness.ProjectedCleanSoundness | |
| open scoped BigOperators | |
| open UniqueGamesTheorem.Foundations.Games | |
| open ProjectedCardinality ProjectedCleanLaw UniformCleanSources UniformCleanSoundness | |
| open OddLists | |
| noncomputable section | |
| attribute [local instance] Classical.propDecidable | |
| variable {I E : Type*} [Fintype I] [DecidableEq I] [Fintype E] [DecidableEq E] | |
| {R J D : I → Type*} | |
| [∀ i, Fintype (R i)] [∀ i, Fintype (J i)] [∀ i, Fintype (D i)] | |
| [∀ i, DecidableEq (R i)] [∀ i, DecidableEq (J i)] [∀ i, DecidableEq (D i)] | |
| (branch : I → Nat → Nat) (t : I → Nat) | |
| (height : (i : I) → J i → Nat) (square : (i : I) → J i → Bool) | |
| (ids : (i : I) → (E × R i) → (j : J i) → | |
| RecursiveSpaces.Slots (branch i) (height i j) → Fin (t i) → Nat) | |
| local notation "Ωtree" => treeFactors branch t height square ids | |
| local notation "ztree" => treeZeros branch t height square ids | |
| variable {Q₁ Q₂ A₁ A₂ : Type*} | |
| [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂] | |
| [DecidableEq Q₁] [DecidableEq Q₂] [DecidableEq A₁] [DecidableEq A₂] | |
| [Nonempty A₁] [Nonempty A₂] | |
| omit [∀ index, DecidableEq (D index)] in | |
| theorem odd_list_bound {s L : Nat} [NeZero s] | |
| (μ : FiniteDistribution E) (left : E → Q₁) (right : E → Q₂) | |
| (projection : E → A₁ → A₂) | |
| (determined : (OddListGame.base μ left right projection).EndpointDetermined) | |
| (gap : (s : ℝ) ^ 2 * (OddListGame.base μ left right projection).value ≤ 1 / 2) | |
| (hL : 0 < L) | |
| (alphabet : UniqueGamesTheorem.Foundations.Repetition.logTwo | |
| ((Fintype.card (OddList s A₁) : ℝ) * (Fintype.card (OddList s A₂) : ℝ)) ≤ L) | |
| (ν : (i : I) → FiniteDistribution (R i)) | |
| (flag : (i : I) → FiniteDistribution Bool) | |
| (nonprojected : (i : I) → E × R i → FiniteDistribution (D i)) | |
| (event : Sample (D := D) Ωtree → Bool) | |
| (strategies : (record : FullRecord (D := D) Ωtree) → | |
| Strategy (CleanIndices Ωtree record → Q₁) (CleanIndices Ωtree record → Q₂) | |
| (CleanIndices Ωtree record → OddList s A₁) | |
| (CleanIndices Ωtree record → OddList s A₂)) | |
| (inclusion : ∀ record x, recordEvent (D := D) Ωtree ztree record x = true → | |
| event x = true → | |
| (IndexedRepetition.game (OddListGame.game s μ left right projection) | |
| (CleanIndices Ωtree record)).wins (strategies record) | |
| (fun i => (x i.1).1) = true) : | |
| (childrenLaw Ωtree (fun _ => μ) ν flag nonprojected).probability event ≤ | |
| ∏ i : I, (1 - (1 - (RepetitionRate.halfRate L : ℝ)) * | |
| ((flag i).weight true * ∏ j : J i, | |
| 1 / (shapeCardinality (branch i) (t i) (height i) (square i) j : ℝ))) := by | |
| apply source_soundness Ωtree (OddListGame.game s μ left right projection) | |
| ν flag nonprojected ztree | |
| (fun i => shapeCardinality (branch i) (t i) (height i) (square i)) _ | |
| event (RepetitionRate.halfRate L : ℝ) _ strategies inclusion | |
| · intro i e j | |
| exact factor_card_eq (square i j) (ids i e j) (fun _ _ => 0) | |
| · intro n | |
| exact OddListGame.repetition_halfRate μ left right projection determined gap hL alphabet n | |
| end | |
| end PerfectCompleteness.ProjectedCleanSoundness | |
| end | |
| end OAI | |