openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Repetition /CleanPhysicalReplay.lean
| import OAI.Computability.PerfectCompleteness.Construction.SourceChildKernelLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.CutSlotOutside | |
| import OAI.Computability.PerfectCompleteness.Foundations.WholeCutExteriorTransportLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.WholeCutReplayGroupingTransport | |
| import OAI.Computability.PerfectCompleteness.Foundations.WholeCutReplayLemmas | |
| import OAI.Computability.PerfectCompleteness.Repetition.CleanEndpointSlots | |
| import OAI.Computability.PerfectCompleteness.Repetition.CleanRecordChildBlocks | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CleanPhysicalReplay | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| noncomputable section | |
| variable {branch : Nat → Nat} {root h t : Nat} | |
| (rows repeats : Nat → Nat) (p : Path branch root (h + 1)) | |
| (outside : Slots branch root → Fin t → MixedSupport.Slot) | |
| (placeholder : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| abbrev Exterior := WholeCutGrouping.Exterior rows repeats p | |
| (CutSlotAssembly.fill p outside placeholder) | |
| def exteriorAt (inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (external : 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) external | |
| theorem cutSlots_fill (inside : Slots branch (h + 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 | |
| theorem childSlots_fill (inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (i : Fin (branch h)) : | |
| childSlots (WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) i = | |
| childSlots inside i := | |
| congrArg (fun ss => childSlots ss i) (cutSlots_fill p outside inside) | |
| def fillChild (inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (i : Fin (branch h)) | |
| (block : CutChildGrouping.Child (C := WholeCutCalls.Index rows repeats p) inside rows i) : | |
| CutChildGrouping.Child (C := WholeCutCalls.Index rows repeats p) | |
| (WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows i := | |
| cast (congrArg (SourceChildKernel.Block (WholeCutCalls.Index rows repeats p) rows) | |
| (childSlots_fill p outside inside i).symm) block | |
| theorem fillChild_heq (inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (i : Fin (branch h)) | |
| (block : CutChildGrouping.Child (C := WholeCutCalls.Index rows repeats p) inside rows i) : | |
| HEq (fillChild rows repeats p outside inside i block) block := | |
| cast_heq _ _ | |
| private theorem cast_zeroBlock {n : Nat} {C : Type*} [Fintype C] | |
| (slots target : Slots branch n → Fin t → MixedSupport.Slot) (hs : slots = target) : | |
| cast (congrArg (SourceChildKernel.Block C rows) hs) | |
| (SourceChildKernel.zeroBlock (C := C) rows slots) = | |
| SourceChildKernel.zeroBlock rows target := by | |
| cases hs | |
| rfl | |
| theorem fillChild_zero (inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (i : Fin (branch h)) : | |
| fillChild rows repeats p outside inside i | |
| (SourceChildKernel.zeroBlock rows (childSlots inside i)) = | |
| SourceChildKernel.zeroBlock rows | |
| (childSlots (WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) i) := | |
| cast_zeroBlock rows _ _ (childSlots_fill p outside inside i).symm | |
| def retainedAt (inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (clean : Fin (branch h) → Prop) | |
| (children : (i : {i : Fin (branch h) // ¬ clean i}) → | |
| CutChildGrouping.Child (C := WholeCutCalls.Index rows repeats p) inside rows i.val) : | |
| WholeCutReplayGrouping.RetainedChildren rows repeats p | |
| (CutSlotAssembly.fill p outside inside) clean := | |
| fun i => fillChild rows repeats p outside inside i.val (children i) | |
| def assembleAt (inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (clean : Fin (branch h) → Prop) | |
| (external : Exterior rows repeats p outside placeholder) | |
| (children : (i : {i : Fin (branch h) // ¬ clean i}) → | |
| CutChildGrouping.Child (C := WholeCutCalls.Index rows repeats p) inside rows i.val) : | |
| WholeCutReplay.Tape rows repeats p (CutSlotAssembly.fill p outside inside) clean := | |
| WholeCutReplayGrouping.assemble rows repeats p (CutSlotAssembly.fill p outside inside) clean | |
| (exteriorAt rows repeats p outside placeholder inside external) | |
| (retainedAt rows repeats p outside inside clean children) | |
| def physicalTape (inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (external : Exterior rows repeats p outside placeholder) | |
| (children : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) : | |
| WholeCutSampler.Tape rows repeats p (CutSlotAssembly.fill p outside inside) := | |
| (WholeCutGrouping.splitTape rows repeats p (CutSlotAssembly.fill p outside inside)).symm | |
| (exteriorAt rows repeats p outside placeholder inside external, | |
| fun i => fillChild rows repeats p outside inside i (children i)) | |
| theorem assembleAt_eq_erase (inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (clean : Fin (branch h) → Prop) | |
| (external : Exterior rows repeats p outside placeholder) | |
| (children : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) : | |
| assembleAt rows repeats p outside placeholder inside clean external (fun i => children i.val) = | |
| WholeCutReplay.erase rows repeats p (CutSlotAssembly.fill p outside inside) clean | |
| (physicalTape rows repeats p outside placeholder inside external children) := by | |
| have h := WholeCutReplayGrouping.assemble_splitTape rows repeats p | |
| (CutSlotAssembly.fill p outside inside) clean | |
| (physicalTape rows repeats p outside placeholder inside external children) | |
| change WholeCutReplayGrouping.assemble rows repeats p (CutSlotAssembly.fill p outside inside) clean | |
| (exteriorAt rows repeats p outside placeholder inside external) | |
| (fun child => fillChild rows repeats p outside inside child.val (children child.val)) = | |
| WholeCutReplay.erase rows repeats p (CutSlotAssembly.fill p outside inside) clean | |
| (physicalTape rows repeats p outside placeholder inside external children) | |
| simpa only [physicalTape, Equiv.apply_symm_apply] using h | |
| theorem physicalTape_grouped_zero | |
| (inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (clean : Fin (branch h) → Prop) | |
| (external : Exterior rows repeats p outside placeholder) | |
| (children : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) | |
| (hzero : ∀ i, clean i → children i = | |
| SourceChildKernel.zeroBlock rows (childSlots inside i)) : | |
| WholeCutZero.GroupedAtClean rows repeats p (CutSlotAssembly.fill p outside inside) clean | |
| (physicalTape rows repeats p outside placeholder inside external children) := by | |
| intro i hi | |
| have hz : fillChild rows repeats p outside inside i (children i) = | |
| SourceChildKernel.zeroBlock rows | |
| (childSlots (WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) i) := by | |
| rw [hzero i hi] | |
| exact fillChild_zero rows repeats p outside inside i | |
| change (∀ q, ((WholeCutGrouping.splitTape rows repeats p | |
| (CutSlotAssembly.fill p outside inside) | |
| (physicalTape rows repeats p outside placeholder inside external children)).2 i).1 q = 0) ∧ _ | |
| simp only [physicalTape, Equiv.apply_symm_apply] | |
| rw [hz] | |
| exact ⟨fun _ => rfl, fun _ _ => rfl⟩ | |
| private theorem exteriorAt_heq | |
| (inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (external : Exterior rows repeats p outside placeholder) : | |
| HEq (exteriorAt rows repeats p outside placeholder inside external) external := | |
| cast_heq _ _ | |
| theorem transport_assembleAt | |
| (inside target : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (clean : Fin (branch h) → Prop) | |
| (hs : ∀ i, ¬ clean i → ∀ s, inside (i, s) = target (i, s)) | |
| (external : Exterior rows repeats p outside placeholder) | |
| (children : (i : {i : Fin (branch h) // ¬ clean i}) → | |
| CutChildGrouping.Child (C := WholeCutCalls.Index rows repeats p) inside rows i.val) | |
| (other : (i : {i : Fin (branch h) // ¬ clean i}) → | |
| CutChildGrouping.Child (C := WholeCutCalls.Index rows repeats p) target rows i.val) | |
| (hc : ∀ i, HEq (children i) (other i)) : | |
| WholeCutReplayKeys.transportTape rows repeats p | |
| (CutSlotAssembly.fill p outside inside) (CutSlotAssembly.fill p outside target) clean | |
| (CutSlotAssembly.fill_kept_eq p outside inside target clean hs) | |
| (assembleAt rows repeats p outside placeholder inside clean external children) = | |
| assembleAt rows repeats p outside placeholder target clean external other := by | |
| unfold assembleAt | |
| rw [WholeCutReplayGroupingTransport.transport_assemble] | |
| apply congrArg₂ (WholeCutReplayGrouping.assemble rows repeats p | |
| (CutSlotAssembly.fill p outside target) clean) | |
| · apply eq_of_heq | |
| exact (cast_heq _ _).trans | |
| ((exteriorAt_heq rows repeats p outside placeholder inside external).trans | |
| (exteriorAt_heq rows repeats p outside placeholder target external).symm) | |
| · funext i | |
| apply eq_of_heq | |
| exact (WholeCutReplayGroupingTransport.castChildren_apply_heq rows repeats p | |
| (CutSlotAssembly.fill p outside inside) (CutSlotAssembly.fill p outside target) clean | |
| (CutSlotAssembly.fill_kept_eq p outside inside target clean hs) | |
| (retainedAt rows repeats p outside inside clean children) i).trans | |
| ((fillChild_heq rows repeats p outside inside i.val (children i)).trans | |
| ((hc i).trans (fillChild_heq rows repeats p outside target i.val (other i)).symm)) | |
| section Source | |
| variable {v m : Nat} [NeZero m] | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch h) → Slots branch h) | |
| abbrev Record := CleanRecordChildBlocks.Record | |
| (C := WholeCutCalls.Index rows repeats p) (t := t) rows clauses designated | |
| abbrev Sample := SourceChildKernel.Sample | |
| (C := WholeCutCalls.Index rows repeats p) (t := t) rows clauses designated | |
| def leftRecord (record : Record (t := t) rows repeats p clauses designated) | |
| (external : Exterior rows repeats p outside placeholder) : | |
| WholeCutReplay.Tape rows repeats p | |
| (CutSlotAssembly.fill p outside (CleanRecordChildBlocks.leftRef rows clauses designated record)) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) := | |
| assembleAt rows repeats p outside placeholder | |
| (CleanRecordChildBlocks.leftRef rows clauses designated record) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) external | |
| (CleanRecordChildBlocks.leftChild rows clauses designated record) | |
| def rightRecord (record : Record (t := t) rows repeats p clauses designated) | |
| (external : Exterior rows repeats p outside placeholder) : | |
| WholeCutReplay.Tape rows repeats p | |
| (CutSlotAssembly.fill p outside (CleanRecordChildBlocks.rightRef rows clauses designated record)) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) := | |
| assembleAt rows repeats p outside placeholder | |
| (CleanRecordChildBlocks.rightRef rows clauses designated record) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) external | |
| (CleanRecordChildBlocks.rightChild rows clauses designated record) | |
| def leftInside (x : Sample (t := t) rows repeats p clauses designated) : | |
| Slots branch (h + 1) → Fin t → MixedSupport.Slot := | |
| SourceChildKernel.parentLeftSlots clauses designated (fun i => ((x i).1, (x i).2.1)) | |
| def rightInside (x : Sample (t := t) rows repeats p clauses designated) : | |
| Slots branch (h + 1) → Fin t → MixedSupport.Slot := | |
| SourceChildKernel.parentRightSlots rows clauses designated | |
| (fun i => ((x i).1, (x i).2.1)) (fun i => (x i).2.2) | |
| def leftBlocks (x : Sample (t := t) rows repeats p clauses designated) : | |
| CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) | |
| (leftInside rows repeats p clauses designated x) rows := | |
| SourceChildKernel.leftChildren rows clauses designated | |
| (fun i => ((x i).1, (x i).2.1)) (fun i => (x i).2.2) | |
| def rightBlocks (x : Sample (t := t) rows repeats p clauses designated) : | |
| CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) | |
| (rightInside rows repeats p clauses designated x) rows := | |
| SourceChildKernel.rightChildren rows clauses designated | |
| (fun i => ((x i).1, (x i).2.1)) (fun i => (x i).2.2) | |
| def leftTape (x : Sample (t := t) rows repeats p clauses designated) | |
| (external : Exterior rows repeats p outside placeholder) : | |
| WholeCutSampler.Tape rows repeats p | |
| (CutSlotAssembly.fill p outside (leftInside rows repeats p clauses designated x)) := | |
| physicalTape rows repeats p outside placeholder (leftInside rows repeats p clauses designated x) | |
| external (leftBlocks rows repeats p clauses designated x) | |
| def rightTape (x : Sample (t := t) rows repeats p clauses designated) | |
| (external : Exterior rows repeats p outside placeholder) : | |
| WholeCutSampler.Tape rows repeats p | |
| (CutSlotAssembly.fill p outside (rightInside rows repeats p clauses designated x)) := | |
| physicalTape rows repeats p outside placeholder (rightInside rows repeats p clauses designated x) | |
| external (rightBlocks rows repeats p clauses designated x) | |
| theorem leftInside_nonclean (record : Record (t := t) rows repeats p clauses designated) | |
| (x : Sample (t := t) rows repeats p clauses designated) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (i : Fin (branch h)) (hi : ¬ CleanRecordChildBlocks.clean rows clauses designated record i) | |
| (s : Slots branch h) : | |
| CleanRecordChildBlocks.leftRef rows clauses designated record (i, s) = | |
| leftInside rows repeats p clauses designated x (i, s) := | |
| congrFun (CleanRecordChildBlocks.leftRef_childSlots_event | |
| rows clauses designated record x hevent ⟨i, hi⟩) s | |
| theorem rightInside_nonclean (record : Record (t := t) rows repeats p clauses designated) | |
| (x : Sample (t := t) rows repeats p clauses designated) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (i : Fin (branch h)) (hi : ¬ CleanRecordChildBlocks.clean rows clauses designated record i) | |
| (s : Slots branch h) : | |
| CleanRecordChildBlocks.rightRef rows clauses designated record (i, s) = | |
| rightInside rows repeats p clauses designated x (i, s) := | |
| congrFun (CleanRecordChildBlocks.rightRef_childSlots_event | |
| rows clauses designated record x hevent ⟨i, hi⟩) s | |
| theorem leftRecord_eq_erase (record : Record (t := t) rows repeats p clauses designated) | |
| (x : Sample (t := t) rows repeats p clauses designated) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (external : Exterior rows repeats p outside placeholder) : | |
| WholeCutReplayKeys.transportTape rows repeats p | |
| (CutSlotAssembly.fill p outside (CleanRecordChildBlocks.leftRef rows clauses designated record)) | |
| (CutSlotAssembly.fill p outside (leftInside rows repeats p clauses designated x)) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (CutSlotAssembly.fill_kept_eq p outside _ _ _ | |
| (leftInside_nonclean rows repeats p clauses designated record x hevent)) | |
| (leftRecord rows repeats p outside placeholder clauses designated record external) = | |
| WholeCutReplay.erase rows repeats p | |
| (CutSlotAssembly.fill p outside (leftInside rows repeats p clauses designated x)) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (leftTape rows repeats p outside placeholder clauses designated x external) := by | |
| exact (transport_assembleAt rows repeats p outside placeholder _ _ _ | |
| (leftInside_nonclean rows repeats p clauses designated record x hevent) external _ _ | |
| (CleanRecordChildBlocks.leftChild_event_heq rows clauses designated record x hevent)).trans | |
| (assembleAt_eq_erase rows repeats p outside placeholder _ _ external | |
| (leftBlocks rows repeats p clauses designated x)) | |
| theorem rightRecord_eq_erase (record : Record (t := t) rows repeats p clauses designated) | |
| (x : Sample (t := t) rows repeats p clauses designated) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (external : Exterior rows repeats p outside placeholder) : | |
| WholeCutReplayKeys.transportTape rows repeats p | |
| (CutSlotAssembly.fill p outside (CleanRecordChildBlocks.rightRef rows clauses designated record)) | |
| (CutSlotAssembly.fill p outside (rightInside rows repeats p clauses designated x)) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (CutSlotAssembly.fill_kept_eq p outside _ _ _ | |
| (rightInside_nonclean rows repeats p clauses designated record x hevent)) | |
| (rightRecord rows repeats p outside placeholder clauses designated record external) = | |
| WholeCutReplay.erase rows repeats p | |
| (CutSlotAssembly.fill p outside (rightInside rows repeats p clauses designated x)) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (rightTape rows repeats p outside placeholder clauses designated x external) := by | |
| exact (transport_assembleAt rows repeats p outside placeholder _ _ _ | |
| (rightInside_nonclean rows repeats p clauses designated record x hevent) external _ _ | |
| (CleanRecordChildBlocks.rightChild_event_heq rows clauses designated record x hevent)).trans | |
| (assembleAt_eq_erase rows repeats p outside placeholder _ _ external | |
| (rightBlocks rows repeats p clauses designated x)) | |
| variable | |
| (μ : Fin (branch h) → FiniteDistribution (SourceQuestionReconstruction.SourceTuple m t)) | |
| (ν : (i : Fin (branch h)) → FiniteDistribution | |
| (SourceChildKernel.Rest (m := m) (t := t) designated i)) | |
| (flag : Fin (branch h) → FiniteDistribution Bool) | |
| omit [NeZero m] in | |
| theorem rightBlocks_zero (record : Record (t := t) rows repeats p clauses designated) | |
| (x : Sample (t := t) rows repeats p clauses designated) | |
| (hx : (SourceChildKernel.childrenLaw rows clauses designated μ ν flag).weight x ≠ 0) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (i : Fin (branch h)) (hi : CleanRecordChildBlocks.clean rows clauses designated record i) : | |
| rightBlocks rows repeats p clauses designated x i = | |
| SourceChildKernel.zeroBlock rows | |
| (childSlots (rightInside rows repeats p clauses designated x) i) := | |
| SourceChildKernel.childrenLaw_clean_zero rows clauses designated μ ν flag x hx i | |
| (CleanRecordChildBlocks.childClean_of_clean rows clauses designated record x hevent i hi) | |
| omit [NeZero m] in | |
| theorem leftBlocks_zero (record : Record (t := t) rows repeats p clauses designated) | |
| (x : Sample (t := t) rows repeats p clauses designated) | |
| (hx : (SourceChildKernel.childrenLaw rows clauses designated μ ν flag).weight x ≠ 0) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (i : Fin (branch h)) (hi : CleanRecordChildBlocks.clean rows clauses designated record i) : | |
| leftBlocks rows repeats p clauses designated x i = | |
| SourceChildKernel.zeroBlock rows | |
| (childSlots (leftInside rows repeats p clauses designated x) i) := by | |
| have hz := rightBlocks_zero rows repeats p clauses designated μ ν flag record x hx hevent i hi | |
| change ChildAssemblyProjection.childPullback rows | |
| (SourceChildKernel.parentProjection rows clauses designated | |
| (fun j => ((x j).1, (x j).2.1)) (fun j => (x j).2.2)) i | |
| (rightBlocks rows repeats p clauses designated x i) = _ | |
| rw [hz] | |
| exact map_zero _ | |
| omit [NeZero m] in | |
| theorem rightTape_grouped_zero (record : Record (t := t) rows repeats p clauses designated) | |
| (x : Sample (t := t) rows repeats p clauses designated) | |
| (hx : (SourceChildKernel.childrenLaw rows clauses designated μ ν flag).weight x ≠ 0) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (external : Exterior rows repeats p outside placeholder) : | |
| WholeCutZero.GroupedAtClean rows repeats p | |
| (CutSlotAssembly.fill p outside (rightInside rows repeats p clauses designated x)) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (rightTape rows repeats p outside placeholder clauses designated x external) := | |
| physicalTape_grouped_zero rows repeats p outside placeholder _ _ external _ | |
| (rightBlocks_zero rows repeats p clauses designated μ ν flag record x hx hevent) | |
| omit [NeZero m] in | |
| theorem leftTape_grouped_zero (record : Record (t := t) rows repeats p clauses designated) | |
| (x : Sample (t := t) rows repeats p clauses designated) | |
| (hx : (SourceChildKernel.childrenLaw rows clauses designated μ ν flag).weight x ≠ 0) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (external : Exterior rows repeats p outside placeholder) : | |
| WholeCutZero.GroupedAtClean rows repeats p | |
| (CutSlotAssembly.fill p outside (leftInside rows repeats p clauses designated x)) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (leftTape rows repeats p outside placeholder clauses designated x external) := | |
| physicalTape_grouped_zero rows repeats p outside placeholder _ _ external _ | |
| (leftBlocks_zero rows repeats p clauses designated μ ν flag record x hx hevent) | |
| private theorem response_of_erase {Y : Type*} | |
| (slots target : Slots branch root → Fin t → MixedSupport.Slot) | |
| (clean : Fin (branch h) → Prop) | |
| (hs : ∀ s, CutSamplerKeyLocality.keptLeaf p clean s → slots s = target s) | |
| (reference : WholeCutReplay.Tape rows repeats p slots clean) | |
| (tape : WholeCutSampler.Tape rows repeats p target) | |
| (htape : WholeCutReplayKeys.transportTape rows repeats p slots target clean hs reference = | |
| WholeCutReplay.erase rows repeats p target clean tape) | |
| (hzero : WholeCutZero.GroupedAtClean rows repeats p target clean tape) | |
| (post : Output branch root rows → Y) | |
| (labeling : KeyStrategy.Strategy (TreeCanonical.locationCount branch root t)) | |
| (side : CanonicalKeys.Side) : | |
| KeyStrategy.response labeling side (TreeCanonical.numberedSlots slots) | |
| (WholeCutReplayKeys.query rows repeats p slots clean reference post) = | |
| KeyStrategy.response labeling side (TreeCanonical.numberedSlots target) | |
| (WholeCutKeyLocality.query rows repeats p target tape post) := by | |
| have h := WholeCutReplayKeys.response_eq_transport rows repeats p slots target clean hs | |
| reference post labeling side | |
| rw [htape, WholeCutReplayKeys.query_erase rows repeats p target clean tape post hzero] at h | |
| exact h | |
| theorem leftRecord_response_eq {Y : Type*} | |
| (record : Record (t := t) rows repeats p clauses designated) | |
| (x : Sample (t := t) rows repeats p clauses designated) | |
| (hx : (SourceChildKernel.childrenLaw rows clauses designated μ ν flag).weight x ≠ 0) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (external : Exterior rows repeats p outside placeholder) | |
| (post : Output branch root rows → Y) | |
| (labeling : KeyStrategy.Strategy (TreeCanonical.locationCount branch root t)) | |
| (side : CanonicalKeys.Side) : | |
| KeyStrategy.response labeling side | |
| (TreeCanonical.numberedSlots | |
| (CutSlotAssembly.fill p outside (CleanRecordChildBlocks.leftRef rows clauses designated record))) | |
| (WholeCutReplayKeys.query rows repeats p | |
| (CutSlotAssembly.fill p outside (CleanRecordChildBlocks.leftRef rows clauses designated record)) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (leftRecord rows repeats p outside placeholder clauses designated record external) post) = | |
| KeyStrategy.response labeling side | |
| (TreeCanonical.numberedSlots | |
| (CutSlotAssembly.fill p outside (leftInside rows repeats p clauses designated x))) | |
| (WholeCutKeyLocality.query rows repeats p | |
| (CutSlotAssembly.fill p outside (leftInside rows repeats p clauses designated x)) | |
| (leftTape rows repeats p outside placeholder clauses designated x external) post) := | |
| response_of_erase rows repeats p _ _ _ | |
| (CutSlotAssembly.fill_kept_eq p outside _ _ _ | |
| (leftInside_nonclean rows repeats p clauses designated record x hevent)) _ _ | |
| (leftRecord_eq_erase rows repeats p outside placeholder clauses designated record x hevent external) | |
| (leftTape_grouped_zero rows repeats p outside placeholder clauses designated μ ν flag | |
| record x hx hevent external) post labeling side | |
| theorem rightRecord_response_eq {Y : Type*} | |
| (record : Record (t := t) rows repeats p clauses designated) | |
| (x : Sample (t := t) rows repeats p clauses designated) | |
| (hx : (SourceChildKernel.childrenLaw rows clauses designated μ ν flag).weight x ≠ 0) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (external : Exterior rows repeats p outside placeholder) | |
| (post : Output branch root rows → Y) | |
| (labeling : KeyStrategy.Strategy (TreeCanonical.locationCount branch root t)) | |
| (side : CanonicalKeys.Side) : | |
| KeyStrategy.response labeling side | |
| (TreeCanonical.numberedSlots | |
| (CutSlotAssembly.fill p outside (CleanRecordChildBlocks.rightRef rows clauses designated record))) | |
| (WholeCutReplayKeys.query rows repeats p | |
| (CutSlotAssembly.fill p outside (CleanRecordChildBlocks.rightRef rows clauses designated record)) | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (rightRecord rows repeats p outside placeholder clauses designated record external) post) = | |
| KeyStrategy.response labeling side | |
| (TreeCanonical.numberedSlots | |
| (CutSlotAssembly.fill p outside (rightInside rows repeats p clauses designated x))) | |
| (WholeCutKeyLocality.query rows repeats p | |
| (CutSlotAssembly.fill p outside (rightInside rows repeats p clauses designated x)) | |
| (rightTape rows repeats p outside placeholder clauses designated x external) post) := | |
| response_of_erase rows repeats p _ _ _ | |
| (CutSlotAssembly.fill_kept_eq p outside _ _ _ | |
| (rightInside_nonclean rows repeats p clauses designated record x hevent)) _ _ | |
| (rightRecord_eq_erase rows repeats p outside placeholder clauses designated record x hevent external) | |
| (rightTape_grouped_zero rows repeats p outside placeholder clauses designated μ ν flag | |
| record x hx hevent external) post labeling side | |
| end Source | |
| section OwnReplay | |
| variable {v m : Nat} [NeZero m] | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch h) → Slots branch h) | |
| (record : Record (t := t) rows repeats p clauses designated) | |
| (external : Exterior rows repeats p outside placeholder) | |
| def leftReplay | |
| (own : {i : Fin (branch h) // CleanRecordChildBlocks.clean rows clauses designated record i} → | |
| Fin t → Fin m) := | |
| CleanEndpointSlots.leftReplay clauses designated | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (CleanRecordChildBlocks.visible rows clauses designated record) rows repeats p outside | |
| (leftRecord rows repeats p outside placeholder clauses designated record external) own | |
| def rightReplay | |
| (own : {i : Fin (branch h) // CleanRecordChildBlocks.clean rows clauses designated record i} → | |
| Fin t → Fin v) := | |
| CleanEndpointSlots.rightReplay clauses designated | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (CleanRecordChildBlocks.visible rows clauses designated record) rows repeats p outside | |
| (CleanRecordChildBlocks.projected rows clauses designated record) | |
| (rightRecord rows repeats p outside placeholder clauses designated record external) own | |
| end OwnReplay | |
| end | |
| end PerfectCompleteness.CleanPhysicalReplay | |
| end | |
| end OAI | |