openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Foundations /CanonicalDirections.lean
| import OAI.Computability.PerfectCompleteness.Foundations.PrefixTests | |
| import OAI.Computability.PerfectCompleteness.Reduction.PreliminaryLevelPair | |
| import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerReplayTransportLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.RationalFiniteLawLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CanonicalDirections | |
| open RecursiveSpaces HierarchicalArrays PreliminarySampler | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n : Nat} | |
| abbrev Tuple (rows : Nat → Nat) (n : Nat) := | |
| (level : Fin n) → PrefixTests.LevelDirection rows level | |
| def directionEquiv (rows : Nat → Nat) (leaf : Slots branch n) (level : Fin n) : | |
| PrefixTests.LevelDirection rows level ≃ | |
| BucketSampler.Direction (rows (Nodes.height (GeometricPath.nodeAtLevel leaf level))) := | |
| (Equiv.cast (congrArg (fun height => BucketSampler.Direction (rows height)) | |
| (GeometricPath.nodeAtLevel_height leaf level))).symm | |
| def tupleEquiv (rows : Nat → Nat) (leaf : Slots branch n) : | |
| Tuple rows n ≃ CandidateCoupling.DirectionTuple rows leaf := | |
| Equiv.piCongrRight (directionEquiv rows leaf) | |
| @[simp] theorem tupleEquiv_apply (rows : Nat → Nat) (leaf : Slots branch n) | |
| (directions : Tuple rows n) (level : Fin n) : | |
| tupleEquiv rows leaf directions level = directionEquiv rows leaf level (directions level) := rfl | |
| @[simp] theorem tupleEquiv_symm_apply (rows : Nat → Nat) (leaf : Slots branch n) | |
| (directions : CandidateCoupling.DirectionTuple rows leaf) (level : Fin n) : | |
| (tupleEquiv rows leaf).symm directions level = | |
| (directionEquiv rows leaf level).symm (directions level) := rfl | |
| def levelLaw (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1)) (level : Fin n) : | |
| FiniteDistribution (PrefixTests.LevelDirection rows level) := by | |
| letI := directionNonempty (rows (level.val + 1)) (hrows level.val) | |
| exact FiniteDistribution.uniform _ | |
| def law (rows : Nat → Nat) (n : Nat) (hrows : ∀ k, 0 < rows (k + 1)) : | |
| FiniteDistribution (Tuple rows n) := | |
| FiniteProduct.law (levelLaw rows hrows) | |
| theorem directionLaw_toCanonical (rows : Nat → Nat) (leaf : Slots branch n) | |
| (level : Fin n) (hrows : ∀ k, 0 < rows (k + 1)) : | |
| (PreliminaryAveraging.directionLaw leaf level hrows).pushforward | |
| (directionEquiv rows leaf level).symm = levelLaw rows hrows level := by | |
| let : Nonempty (PrefixTests.LevelDirection rows level) := | |
| directionNonempty (rows (level.val + 1)) (hrows level.val) | |
| let : Nonempty (BucketSampler.Direction | |
| (rows (Nodes.height (GeometricPath.nodeAtLevel leaf level)))) := | |
| directionNonempty _ (by rw [GeometricPath.nodeAtLevel_height]; exact hrows level.val) | |
| change (FiniteDistribution.uniform | |
| (BucketSampler.Direction (rows (Nodes.height (GeometricPath.nodeAtLevel leaf level))))).pushforward | |
| (directionEquiv rows leaf level).symm = | |
| FiniteDistribution.uniform (PrefixTests.LevelDirection rows level) | |
| rw [← FiniteDistribution.transport_eq_pushforward] | |
| exact UniformConditioning.uniform_transport (directionEquiv rows leaf level).symm | |
| theorem directionsLaw_toCanonical (rows : Nat → Nat) (leaf : Slots branch n) | |
| (hrows : ∀ k, 0 < rows (k + 1)) : | |
| (CandidateCoupling.directionsLaw leaf hrows).pushforward (tupleEquiv rows leaf).symm = | |
| law rows n hrows := by | |
| change (FiniteProduct.law (fun level => PreliminaryAveraging.directionLaw leaf level hrows)).pushforward | |
| (fun directions level => (directionEquiv rows leaf level).symm (directions level)) = _ | |
| rw [FiniteProduct.pushforward_map] | |
| apply congrArg (fun laws : (level : Fin n) → | |
| FiniteDistribution (PrefixTests.LevelDirection rows level) => FiniteProduct.law laws) | |
| funext level | |
| exact directionLaw_toCanonical rows leaf level hrows | |
| theorem canonicalLaw_toDirections (rows : Nat → Nat) (leaf : Slots branch n) | |
| (hrows : ∀ k, 0 < rows (k + 1)) : | |
| (law rows n hrows).pushforward (tupleEquiv rows leaf) = | |
| CandidateCoupling.directionsLaw leaf hrows := by | |
| have h := congrArg | |
| (fun μ : FiniteDistribution (Tuple rows n) => μ.pushforward (tupleEquiv rows leaf)) | |
| (directionsLaw_toCanonical rows leaf hrows) | |
| rw [FiniteDistribution.pushforward_comp] at h | |
| have hid : (fun directions : CandidateCoupling.DirectionTuple rows leaf => | |
| tupleEquiv rows leaf ((tupleEquiv rows leaf).symm directions)) = id := by | |
| funext directions | |
| exact (tupleEquiv rows leaf).apply_symm_apply directions | |
| rw [hid, FiniteDistribution.pushforward_id] at h | |
| exact h.symm | |
| @[simp] theorem choiceAt_toCanonical (rows : Nat → Nat) (leaf : Slots branch n) | |
| (directions : CandidateCoupling.DirectionTuple rows leaf) (level : Fin n) : | |
| PrefixTests.choiceAt rows leaf level ((tupleEquiv rows leaf).symm directions level) = | |
| (⟨level, directions level⟩ : Choice rows leaf) := by | |
| change (⟨level, directionEquiv rows leaf level | |
| ((directionEquiv rows leaf level).symm (directions level))⟩ : Choice rows leaf) = _ | |
| rw [Equiv.apply_symm_apply] | |
| variable {v m t : Nat} {rows repeats : Nat → Nat} | |
| theorem win_eq_observe_canonical (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (strategy : PreliminaryStrategy.Strategy clauses branch n t rows repeats) | |
| (b : Base branch n t m) | |
| (arrays : Arrays (HierarchicalArrays.sourceSlots clauses (endpoints b.1)) rows) | |
| (directions : Tuple rows n) (level : Fin n) : | |
| PreliminaryLevelPair.win clauses strategy level | |
| ⟨b, arrays, tupleEquiv rows b.2 directions⟩ = | |
| PreliminaryStrategy.observe clauses | |
| (PreliminaryStrategy.extend clauses branch n t rows repeats strategy) b arrays | |
| (PrefixTests.choiceAt rows b.2 level (directions level)) := rfl | |
| theorem win_eq_observe (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (strategy : PreliminaryStrategy.Strategy clauses branch n t rows repeats) | |
| (level : Fin n) (e : CandidateCoupling.Shared clauses branch n t rows) : | |
| PreliminaryLevelPair.win clauses strategy level e = | |
| PreliminaryStrategy.observe clauses | |
| (PreliminaryStrategy.extend clauses branch n t rows repeats strategy) e.1 e.2.1 | |
| (PrefixTests.choiceAt rows e.1.2 level | |
| ((tupleEquiv rows e.1.2).symm e.2.2 level)) := by | |
| rw [choiceAt_toCanonical] | |
| rfl | |
| end | |
| end PerfectCompleteness.CanonicalDirections | |
| end | |
| end OAI | |