openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /FilledUniformSampler.lean
| import OAI.Computability.PerfectCompleteness.Foundations.CutSlotOutside | |
| import OAI.Computability.PerfectCompleteness.Foundations.WholeCutExteriorTransportLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.WholeCutGrouping | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FilledUniformSampler | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| (rows repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (placeholder : Slots branch (m + 1) → Fin t → MixedSupport.Slot) | |
| abbrev Exterior := WholeCutGrouping.Exterior rows repeats p | |
| (CutSlotAssembly.fill p outside placeholder) | |
| theorem cutSlots_fill (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) : | |
| WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside) = inside := by | |
| funext s | |
| exact CutSlotAssembly.fill_at_cut p outside inside s | |
| def exteriorEquiv (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) : | |
| Exterior rows repeats p outside placeholder ≃ | |
| WholeCutGrouping.Exterior rows repeats p (CutSlotAssembly.fill p outside inside) := | |
| WholeCutExteriorTransport.equiv rows repeats p | |
| (CutSlotAssembly.fill p outside placeholder) (CutSlotAssembly.fill p outside inside) | |
| (CutSlotAssembly.fill_outside_eq p outside placeholder inside) | |
| def childrenEquiv (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) : | |
| CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows ≃ | |
| CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) | |
| (WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows := | |
| Equiv.piCongrRight (fun i => Equiv.cast (congrArg | |
| (fun slots => CutChildGrouping.Child (C := WholeCutCalls.Index rows repeats p) slots rows i) | |
| (cutSlots_fill p outside inside).symm)) | |
| @[simp] theorem childrenEquiv_apply | |
| (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) | |
| (children : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) | |
| (i : Fin (branch m)) : | |
| childrenEquiv rows repeats p outside inside children i = | |
| cast (congrArg | |
| (fun slots => CutChildGrouping.Child (C := WholeCutCalls.Index rows repeats p) slots rows i) | |
| (cutSlots_fill p outside inside).symm) (children i) := rfl | |
| theorem childrenEquiv_heq | |
| (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) | |
| (children : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) | |
| (i : Fin (branch m)) : | |
| HEq (childrenEquiv rows repeats p outside inside children i) (children i) := | |
| cast_heq _ _ | |
| def filledSplitEquiv (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) : | |
| (Exterior rows repeats p outside placeholder × | |
| CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) ≃ | |
| WholeCutSampler.Tape rows repeats p (CutSlotAssembly.fill p outside inside) := | |
| (Equiv.prodCongr (exteriorEquiv rows repeats p outside placeholder inside) | |
| (childrenEquiv rows repeats p outside inside)).trans | |
| (WholeCutGrouping.splitTape rows repeats p (CutSlotAssembly.fill p outside inside)).symm | |
| theorem filledSplitEquiv_apply | |
| (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) | |
| (external : Exterior rows repeats p outside placeholder) | |
| (children : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) : | |
| filledSplitEquiv rows repeats p outside placeholder inside (external, children) = | |
| (WholeCutGrouping.splitTape rows repeats p (CutSlotAssembly.fill p outside inside)).symm | |
| (exteriorEquiv rows repeats p outside placeholder inside external, | |
| childrenEquiv rows repeats p outside inside children) := rfl | |
| theorem filledSplitEquiv_tapeLaw | |
| (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) : | |
| ((FiniteDistribution.uniform (Exterior rows repeats p outside placeholder)).product | |
| (CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows)).pushforward | |
| (filledSplitEquiv rows repeats p outside placeholder inside) = | |
| WholeCutSampler.tapeLaw rows repeats p (CutSlotAssembly.fill p outside inside) := by | |
| unfold CutChildGrouping.rawLaw | |
| rw [UniformLinearImage.law_uniform, WholeCutSampler.uniform_product, | |
| WholeCutSampler.tapeLaw_eq_uniform, FiniteDistribution.pushforward_equiv] | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro x | |
| change 1 / (Fintype.card (Exterior rows repeats p outside placeholder × | |
| CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) : ℝ) = | |
| 1 / (Fintype.card | |
| (WholeCutSampler.Tape rows repeats p (CutSlotAssembly.fill p outside inside)) : ℝ) | |
| rw [Fintype.card_congr (filledSplitEquiv rows repeats p outside placeholder inside)] | |
| def evaluate (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) | |
| (x : Exterior rows repeats p outside placeholder × | |
| CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) : | |
| Arrays (CutSlotAssembly.fill p outside inside) rows := | |
| WholeCutSampler.evaluate rows repeats p (CutSlotAssembly.fill p outside inside) | |
| (filledSplitEquiv rows repeats p outside placeholder inside x) | |
| theorem evaluate_law (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) : | |
| ((FiniteDistribution.uniform (Exterior rows repeats p outside placeholder)).product | |
| (CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows)).pushforward | |
| (evaluate rows repeats p outside placeholder inside) = | |
| WholeArraySampler.law rows repeats p (CutSlotAssembly.fill p outside inside) := by | |
| unfold evaluate | |
| rw [← FiniteDistribution.pushforward_comp | |
| ((FiniteDistribution.uniform (Exterior rows repeats p outside placeholder)).product | |
| (CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows)) | |
| (filledSplitEquiv rows repeats p outside placeholder inside) | |
| (WholeCutSampler.evaluate rows repeats p (CutSlotAssembly.fill p outside inside)), | |
| filledSplitEquiv_tapeLaw] | |
| exact WholeCutSampler.evaluate_law rows repeats p (CutSlotAssembly.fill p outside inside) | |
| end | |
| end PerfectCompleteness.FilledUniformSampler | |
| end | |
| end OAI | |