openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Repetition /CleanPhysicalRawChildren.lean
| import OAI.Computability.PerfectCompleteness.Decoding.LowerProjectedOriginalSuccessLemmas | |
| import OAI.Computability.PerfectCompleteness.Repetition.CleanPhysicalReplayLaw | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CleanPhysicalRawChildren | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| 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 inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) | |
| def fillChildren | |
| (children : 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 := | |
| fun i => CleanPhysicalReplay.fillChild rows repeats p outside inside i (children i) | |
| theorem fillChildren_eq_childrenEquiv : | |
| fillChildren rows repeats p outside inside = | |
| (FilledUniformSampler.childrenEquiv rows repeats p outside inside : _ → _) := by | |
| funext children i | |
| exact eq_of_heq ((CleanPhysicalReplay.fillChild_heq rows repeats p outside inside i | |
| (children i)).trans | |
| (FilledUniformSampler.childrenEquiv_heq rows repeats p outside inside children i).symm) | |
| theorem fillChildren_law : | |
| (CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows).pushforward | |
| (fillChildren rows repeats p outside inside) = | |
| CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) | |
| (WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows := by | |
| rw [fillChildren_eq_childrenEquiv] | |
| simp only [CutChildGrouping.rawLaw, UniformLinearImage.law_uniform] | |
| rw [FiniteDistribution.pushforward_equiv] | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro children | |
| change 1 / (Fintype.card | |
| (CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) : ℝ) = | |
| 1 / (Fintype.card (CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) | |
| (WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows) : ℝ) | |
| rw [Fintype.card_congr (FilledUniformSampler.childrenEquiv rows repeats p outside inside)] | |
| theorem expectation_physicalTape | |
| (external : CleanPhysicalReplay.Exterior rows repeats p outside placeholder) | |
| (statistic : WholeCutSampler.Tape rows repeats p (CutSlotAssembly.fill p outside inside) → ℝ) : | |
| (CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows).expectation | |
| (fun children => statistic | |
| (CleanPhysicalReplay.physicalTape rows repeats p outside placeholder inside external children)) = | |
| (CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) | |
| (WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows).expectation | |
| (fun children => statistic | |
| ((WholeCutGrouping.splitTape rows repeats p (CutSlotAssembly.fill p outside inside)).symm | |
| (CleanPhysicalReplay.exteriorAt rows repeats p outside placeholder inside external, | |
| children))) := by | |
| have h := congrArg | |
| (fun law : FiniteDistribution (CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) | |
| (WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows) => | |
| law.expectation (fun children => statistic | |
| ((WholeCutGrouping.splitTape rows repeats p (CutSlotAssembly.fill p outside inside)).symm | |
| (CleanPhysicalReplay.exteriorAt rows repeats p outside placeholder inside external, | |
| children)))) | |
| (fillChildren_law rows repeats p outside inside) | |
| rw [FiniteDistribution.expectation_pushforward] at h | |
| exact h | |
| theorem expectation_lowerTape (upper : Nodes branch n) | |
| (pDown : Path branch (Nodes.height upper) (m + 1)) | |
| (external : CleanPhysicalReplay.Exterior rows repeats | |
| ((Nodes.path upper).append pDown) outside placeholder) | |
| (statistic : WholeCutSampler.Tape rows repeats ((Nodes.path upper).append pDown) | |
| (CutSlotAssembly.fill ((Nodes.path upper).append pDown) outside inside) → ℝ) : | |
| (CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats ((Nodes.path upper).append pDown)) | |
| inside rows).expectation (fun children => statistic | |
| (CleanPhysicalReplay.physicalTape rows repeats ((Nodes.path upper).append pDown) | |
| outside placeholder inside external children)) = | |
| (CutChildGrouping.rawLaw | |
| (C := LowerCutPair.Calls rows repeats ((Nodes.path upper).append pDown)) | |
| (WholeCutGrouping.cutSlots ((Nodes.path upper).append pDown) | |
| (CutSlotAssembly.fill ((Nodes.path upper).append pDown) outside inside)) rows).expectation | |
| (fun children => statistic | |
| (LowerPhysicalFiberSuccess.tape | |
| (CutSlotAssembly.fill ((Nodes.path upper).append pDown) outside inside) upper pDown | |
| (CleanPhysicalReplay.exteriorAt rows repeats ((Nodes.path upper).append pDown) | |
| outside placeholder inside external) children)) := | |
| expectation_physicalTape rows repeats ((Nodes.path upper).append pDown) | |
| outside placeholder inside external statistic | |
| end | |
| end PerfectCompleteness.CleanPhysicalRawChildren | |
| end | |
| end OAI | |