openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /PreliminarySampler.lean
| import OAI.Computability.PerfectCompleteness.Foundations.GeometricPathLemmas | |
| import OAI.Computability.PerfectCompleteness.Reduction.CompletionAlphabetLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.WholeArraySampler | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.PreliminarySampler | |
| open RecursiveSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped Classical | |
| noncomputable section | |
| variable {v m n t : Nat} {branch rows repeats : Nat → Nat} | |
| abbrev Questions (branch : Nat → Nat) (n t m : Nat) := | |
| Slots branch n → Fin t → Fin m | |
| def endpoints (E : Questions branch n t m) : | |
| Slots branch n → Fin t → SourceKeys.Endpoint v m := | |
| fun leaf coordinate => .clause (E leaf coordinate) | |
| abbrev Base (branch : Nat → Nat) (n t m : Nat) := | |
| Questions branch n t m × Slots branch n | |
| abbrev ArrayTape (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (rows repeats : Nat → Nat) (b : Base branch n t m) := | |
| WholeArraySampler.Tape rows repeats (GeometricPath.leafPath b.2) | |
| (sourceSlots clauses (endpoints b.1)) | |
| abbrev Choice (rows : Nat → Nat) (leaf : Slots branch n) := | |
| (level : Fin n) × BucketSampler.Direction | |
| (rows (Nodes.height (GeometricPath.nodeAtLevel leaf level))) | |
| abbrev Raw (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) := | |
| (b : Base branch n t m) × (ArrayTape clauses rows repeats b × Choice rows b.2) | |
| theorem directionNonempty (width : Nat) (positive : 0 < width) : | |
| Nonempty (BucketSampler.Direction width) := by | |
| refine ⟨⟨fun _ => 1, ?_⟩⟩ | |
| intro hz | |
| have h := congrFun hz ⟨0, positive⟩ | |
| change (1 : BucketSampler.F2) = 0 at h | |
| exact one_ne_zero h | |
| def questionsLaw [NeZero m] : FiniteDistribution (Questions branch n t m) := | |
| FiniteDistribution.uniform _ | |
| theorem questionsLaw_eq_product [NeZero m] : | |
| questionsLaw (branch := branch) (n := n) (t := t) (m := m) = | |
| FiniteProduct.law (fun _ : Slots branch n => | |
| FiniteProduct.law (fun _ : Fin t => FiniteDistribution.uniform (Fin m))) := by | |
| simp only [UniformLinearImage.law_uniform] | |
| rfl | |
| def baseLaw [NeZero m] (hbranch : ∀ k < n, 0 < branch k) : | |
| FiniteDistribution (Base branch n t m) := | |
| questionsLaw.product (GeometricPath.law n hbranch) | |
| def choiceLaw (leaf : Slots branch n) (hn : 0 < n) | |
| (hrows : ∀ k, 0 < rows (k + 1)) : FiniteDistribution (Choice rows leaf) := by | |
| letI : Nonempty (Fin n) := ⟨⟨0, hn⟩⟩ | |
| exact CompletionSoundness.sigmaLaw (FiniteDistribution.uniform (Fin n)) fun 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 choiceLaw_level (leaf : Slots branch n) (hn : 0 < n) | |
| (hrows : ∀ k, 0 < rows (k + 1)) (event : Fin n → Bool) : | |
| (choiceLaw leaf hn hrows).probability (fun c => event c.1) = | |
| (by | |
| letI : Nonempty (Fin n) := ⟨⟨0, hn⟩⟩ | |
| exact (FiniteDistribution.uniform (Fin n)).probability event) := by | |
| exact CompletionSoundness.sigmaLaw_probability_first _ _ event | |
| def conditionalLaw (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (rows repeats : Nat → Nat) (hn : 0 < n) | |
| (hrows : ∀ k, 0 < rows (k + 1)) (b : Base branch n t m) : | |
| FiniteDistribution (ArrayTape clauses rows repeats b × Choice rows b.2) := | |
| (WholeArraySampler.tapeLaw rows repeats (GeometricPath.leafPath b.2) | |
| (sourceSlots clauses (endpoints b.1))).product (choiceLaw b.2 hn hrows) | |
| def law [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)) : | |
| FiniteDistribution (Raw clauses branch n t rows repeats) := | |
| CompletionSoundness.sigmaLaw (baseLaw hbranch) | |
| (conditionalLaw clauses rows repeats hn hrows) | |
| theorem law_base [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)) | |
| (event : Base branch n t m → Bool) : | |
| (law clauses rows repeats hn hbranch hrows).probability (fun e => event e.1) = | |
| (baseLaw hbranch).probability event := | |
| CompletionSoundness.sigmaLaw_probability_first _ _ event | |
| theorem law_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)) | |
| (event : Raw clauses branch n t rows repeats → Bool) : | |
| (law clauses rows repeats hn hbranch hrows).probability event = | |
| (baseLaw hbranch).expectation (fun b => | |
| (conditionalLaw clauses rows repeats hn hrows b).probability | |
| (fun tapeAndChoice => event ⟨b, tapeAndChoice⟩)) := | |
| CompletionSoundness.sigmaLaw_probability _ _ event | |
| def family (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) : | |
| HierarchicalGame.Family clauses branch n t rows (Raw clauses branch n t rows repeats) where | |
| endpoints e := endpoints e.1.1 | |
| arrays e := WholeArraySampler.evaluate rows repeats (GeometricPath.leafPath e.1.2) | |
| (sourceSlots clauses (endpoints e.1.1)) e.2.1 | |
| selected e := GeometricPath.nodeAtLevel e.1.2 e.2.2.1 | |
| direction e := e.2.2.2.val | |
| nonzero e := e.2.2.2.property | |
| theorem family_selected_height (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) | |
| (e : Raw clauses branch n t rows repeats) : | |
| Nodes.height ((family clauses branch n t rows repeats).selected e) = e.2.2.1.val + 1 := | |
| GeometricPath.nodeAtLevel_height _ _ | |
| def game [NeZero m] (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) (hn : 0 < n) | |
| (hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) := | |
| HierarchicalGame.game (family clauses branch n t rows repeats) | |
| (law clauses rows repeats hn hbranch hrows) | |
| theorem edge_at_most_two (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) | |
| (e : Raw clauses branch n t rows repeats) | |
| (Q : HierarchicalGame.RightLabel (family clauses branch n t rows repeats) | |
| (HierarchicalGame.rightAt (family clauses branch n t rows repeats) e)) : | |
| Nat.card {P : HierarchicalGame.LeftLabel (family clauses branch n t rows repeats) | |
| (HierarchicalGame.leftAt (family clauses branch n t rows repeats) e) // | |
| HierarchicalGame.edge (family clauses branch n t rows repeats) e P = Q} ≤ 2 := | |
| HierarchicalGame.edge_at_most_two _ e Q | |
| theorem completeness [NeZero m] (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) (hn : 0 < n) | |
| (hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) | |
| (assignment : Fin v → Bool) (hsat : ∀ c, (clauses c).clause.eval assignment = true) : | |
| (game clauses branch n t rows repeats hn hbranch hrows).success | |
| (HierarchicalGame.satisfyingStrategy (family clauses branch n t rows repeats) | |
| assignment hsat) = 1 := | |
| HierarchicalGame.success_satisfyingStrategy _ _ assignment hsat | |
| end | |
| end PerfectCompleteness.PreliminarySampler | |
| end | |
| end OAI | |