openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Repetition /CleanRecordChildBlocks.lean
| import OAI.Computability.PerfectCompleteness.Construction.SourceChildKernelLemmas | |
| import OAI.Computability.PerfectCompleteness.Repetition.CleanEndpointSlots | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CleanRecordChildBlocks | |
| noncomputable section | |
| open scoped Classical | |
| open UniqueGamesTheorem.Foundations.Games | |
| open RecursiveSpaces TreeSourceSpaces HierarchicalArrays | |
| open SourceQuestionReconstruction | |
| variable {branch : Nat → Nat} {h t v m : Nat} {C : Type*} [Fintype C] | |
| (rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch h) → Slots branch h) | |
| abbrev Record := | |
| CleanSourceRecord.Record | |
| (D := SourceChildKernel.D (C := C) (t := t) rows clauses designated) designated | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) | |
| abbrev Sample := SourceChildKernel.Sample (C := C) (t := t) rows clauses designated | |
| abbrev Row (i : Fin (branch h)) := SourceTuple m t × | |
| (SourceChildKernel.Rest (m := m) (t := t) designated i × | |
| SourceChildKernel.Raw (C := C) (t := t) rows clauses designated i) | |
| def clean (record : Record (C := C) (t := t) rows clauses designated) : | |
| Fin (branch h) → Prop := | |
| CleanSourceRecord.clean designated | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) record | |
| def visible (record : Record (C := C) (t := t) rows clauses designated) : | |
| Visible (E := SourceTuple m t) designated (clean rows clauses designated record) := | |
| CleanSourceRecord.visible designated | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) record | |
| def projected (record : Record (C := C) (t := t) rows clauses designated) : | |
| Fin (branch h) → Bool := | |
| CleanSourceRecord.projected designated | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) record | |
| def event (record : Record (C := C) (t := t) rows clauses designated) : | |
| Sample (C := C) (t := t) rows clauses designated → Bool := | |
| UniformCleanSources.recordEvent | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) | |
| (SourceChildKernel.zero (C := C) (t := t) rows clauses designated) record | |
| abbrev Nonclean (record : Record (C := C) (t := t) rows clauses designated) := | |
| {i : Fin (branch h) // ¬ clean rows clauses designated record i} | |
| def extract (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) : Row (C := C) (t := t) rows clauses designated i.val := | |
| match hr : record i.val with | |
| | .inl row => row | |
| | .inr _ => False.elim (i.property (by | |
| simp only [clean, CleanSourceRecord.clean, hr, UniformCleanSources.cleanRecord])) | |
| theorem record_extract (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) : | |
| record i.val = .inl (extract rows clauses designated record i) := by | |
| unfold extract | |
| split | |
| · rename_i row hr | |
| exact hr | |
| · rename_i rest hr | |
| exact False.elim (i.property (by | |
| simp only [clean, CleanSourceRecord.clean, hr, UniformCleanSources.cleanRecord])) | |
| theorem extract_eq (record : Record (C := C) (t := t) rows clauses designated) | |
| (x : Sample (C := C) (t := t) rows clauses designated) | |
| (hevent : event rows clauses designated record x = true) | |
| (i : Nonclean rows clauses designated record) : | |
| extract rows clauses designated record i = x i.val := by | |
| have obs := CleanSourceRecord.observation_eq designated | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) | |
| (SourceChildKernel.zero (C := C) (t := t) rows clauses designated) record x hevent i.val | |
| cases hc : UniformCleanSources.childClean | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) | |
| (SourceChildKernel.zero (C := C) (t := t) rows clauses designated) | |
| i.val (x i.val).1 (x i.val).2.1 (x i.val).2.2 with | |
| | false => | |
| have hr : record i.val = .inl (x i.val) := by | |
| simpa only [SelectedSources.childObservation, hc, Bool.false_eq_true, ite_false] | |
| using obs.symm | |
| exact Sum.inl.inj ((record_extract rows clauses designated record i).symm.trans hr) | |
| | true => | |
| have hr : record i.val = .inr (x i.val).2.1 := by | |
| simpa only [SelectedSources.childObservation, hc, ite_true] using obs.symm | |
| exact False.elim (i.property (by | |
| simp only [clean, CleanSourceRecord.clean, hr, UniformCleanSources.cleanRecord])) | |
| def source (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) : | |
| SourceChildKernel.Source (m := m) (t := t) designated i.val := | |
| ((extract rows clauses designated record i).1, (extract rows clauses designated record i).2.1) | |
| def raw (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) : | |
| SourceChildKernel.Raw (C := C) (t := t) rows clauses designated i.val := | |
| (extract rows clauses designated record i).2.2 | |
| theorem visible_nonclean (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) (leaf : Slots branch h) | |
| (hleaf : ¬ (clean rows clauses designated record i.val ∧ leaf = designated i.val)) : | |
| visible rows clauses designated record i.val ⟨leaf, hleaf⟩ = | |
| SourceChildKernel.tree designated i.val (source rows clauses designated record i) leaf := by | |
| dsimp only [visible, CleanSourceRecord.visible] | |
| split | |
| · rename_i row hr | |
| have hrow : row = extract rows clauses designated record i := | |
| Sum.inl.inj (hr.symm.trans (record_extract rows clauses designated record i)) | |
| rw [hrow] | |
| rfl | |
| · rename_i rest hr | |
| have hbad := hr.symm.trans (record_extract rows clauses designated record i) | |
| cases hbad | |
| theorem projected_nonclean (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) : | |
| projected rows clauses designated record i.val = | |
| SourceChildKernel.rawProjected rows clauses designated i.val | |
| (raw rows clauses designated record i) := by | |
| simp only [projected, CleanSourceRecord.projected, | |
| record_extract rows clauses designated record i, raw, | |
| CleanSourceRecord.rawProjected, SourceChildKernel.rawProjected] | |
| cases (extract rows clauses designated record i).2.2 <;> rfl | |
| variable [NeZero m] | |
| def leftRef (record : Record (C := C) (t := t) rows clauses designated) : | |
| Slots branch (h + 1) → Fin t → MixedSupport.Slot := | |
| CleanEndpointSlots.leftReference clauses designated (clean rows clauses designated record) | |
| (visible rows clauses designated record) | |
| def rightRef (record : Record (C := C) (t := t) rows clauses designated) : | |
| Slots branch (h + 1) → Fin t → MixedSupport.Slot := | |
| CleanEndpointSlots.rightReference clauses designated (clean rows clauses designated record) | |
| (visible rows clauses designated record) (projected rows clauses designated record) | |
| theorem leftRef_childSlots (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) : | |
| childSlots (leftRef rows clauses designated record) i.val = | |
| SourceChildKernel.nativeSlots clauses designated i.val (source rows clauses designated record i) := by | |
| funext leaf k | |
| have hn : ¬ (clean rows clauses designated record i.val ∧ leaf = designated i.val) := | |
| fun h => i.property h.1 | |
| simp only [leftRef, CleanEndpointSlots.leftReference, CleanEndpointSlots.leftInside, | |
| childSlots, reconstructLeft, reconstruct, dite_eq_right hn] | |
| rw [visible_nonclean rows clauses designated record i leaf hn] | |
| rfl | |
| theorem rightRef_childSlots (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) : | |
| childSlots (rightRef rows clauses designated record) i.val = | |
| SourceChildKernel.mixedSlots clauses designated i.val (source rows clauses designated record i) | |
| (SourceChildKernel.rawProjected rows clauses designated i.val (raw rows clauses designated record i)) := by | |
| funext leaf k | |
| have hn : ¬ (clean rows clauses designated record i.val ∧ leaf = designated i.val) := | |
| fun h => i.property h.1 | |
| simp only [rightRef, CleanEndpointSlots.rightReference, CleanEndpointSlots.rightInside, | |
| childSlots, reconstructRight, reconstruct, dite_eq_right hn] | |
| rw [visible_nonclean rows clauses designated record i leaf hn, | |
| projected_nonclean rows clauses designated record i] | |
| exact (SourceChildKernel.mixedSlots_eq_rightTuple clauses designated i.val | |
| (source rows clauses designated record i) | |
| (SourceChildKernel.rawProjected rows clauses designated i.val (raw rows clauses designated record i)) | |
| leaf k).symm | |
| theorem leftRef_childSlots_event (record : Record (C := C) (t := t) rows clauses designated) | |
| (x : Sample (C := C) (t := t) rows clauses designated) | |
| (hevent : event rows clauses designated record x = true) | |
| (i : Nonclean rows clauses designated record) : | |
| childSlots (leftRef rows clauses designated record) i.val = | |
| childSlots (SourceChildKernel.parentLeftSlots clauses designated | |
| (fun j => ((x j).1, (x j).2.1))) i.val := | |
| (leftRef_childSlots rows clauses designated record i).trans | |
| (congrArg (fun y : Row (C := C) (t := t) rows clauses designated i.val => | |
| SourceChildKernel.nativeSlots clauses designated i.val (y.1, y.2.1)) | |
| (extract_eq rows clauses designated record x hevent i)) | |
| theorem rightRef_childSlots_event (record : Record (C := C) (t := t) rows clauses designated) | |
| (x : Sample (C := C) (t := t) rows clauses designated) | |
| (hevent : event rows clauses designated record x = true) | |
| (i : Nonclean rows clauses designated record) : | |
| childSlots (rightRef rows clauses designated record) i.val = | |
| childSlots (SourceChildKernel.parentRightSlots rows clauses designated | |
| (fun j => ((x j).1, (x j).2.1)) (fun j => (x j).2.2)) i.val := | |
| (rightRef_childSlots rows clauses designated record i).trans | |
| (congrArg (fun y : Row (C := C) (t := t) rows clauses designated i.val => | |
| SourceChildKernel.mixedSlots clauses designated i.val (y.1, y.2.1) | |
| (SourceChildKernel.rawProjected rows clauses designated i.val y.2.2)) | |
| (extract_eq rows clauses designated record x hevent i)) | |
| def physicalRight (i : Fin (branch h)) (row : Row (C := C) (t := t) rows clauses designated i) : | |
| SourceChildKernel.Block C rows (SourceChildKernel.mixedSlots clauses designated i (row.1, row.2.1) | |
| (SourceChildKernel.rawProjected rows clauses designated i row.2.2)) := | |
| SourceChildKernel.decodeRight rows clauses designated i (row.1, row.2.1) row.2.2 | |
| def physicalLeft (i : Fin (branch h)) (row : Row (C := C) (t := t) rows clauses designated i) : | |
| SourceChildKernel.Block C rows (SourceChildKernel.nativeSlots clauses designated i (row.1, row.2.1)) := | |
| let p := SourceChildKernel.projection clauses designated i (row.1, row.2.1) | |
| (SourceChildKernel.rawProjected rows clauses designated i row.2.2) | |
| let block := physicalRight rows clauses designated i row | |
| (fun call => ChildBlockProjection.squarePullback p (block.1 call), | |
| ChildBlockProjection.arraysPullback rows p block.2) | |
| omit [NeZero m] in | |
| theorem physicalLeft_eq_leftChildren | |
| (x : Sample (C := C) (t := t) rows clauses designated) (i : Fin (branch h)) : | |
| physicalLeft rows clauses designated i (x i) = | |
| SourceChildKernel.leftChildren rows clauses designated | |
| (fun j => ((x j).1, (x j).2.1)) (fun j => (x j).2.2) i := by | |
| rw [SourceChildKernel.leftChildren_apply] | |
| rfl | |
| def rightChild (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) : | |
| SourceChildKernel.Block C rows (childSlots (rightRef rows clauses designated record) i.val) := | |
| cast (congrArg (SourceChildKernel.Block C rows) (rightRef_childSlots rows clauses designated record i).symm) | |
| (physicalRight rows clauses designated i.val (extract rows clauses designated record i)) | |
| def leftChild (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) : | |
| SourceChildKernel.Block C rows (childSlots (leftRef rows clauses designated record) i.val) := | |
| cast (congrArg (SourceChildKernel.Block C rows) (leftRef_childSlots rows clauses designated record i).symm) | |
| (physicalLeft rows clauses designated i.val (extract rows clauses designated record i)) | |
| theorem rightChild_heq (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) : | |
| HEq (rightChild rows clauses designated record i) | |
| (physicalRight rows clauses designated i.val (extract rows clauses designated record i)) := | |
| cast_heq _ _ | |
| theorem leftChild_heq (record : Record (C := C) (t := t) rows clauses designated) | |
| (i : Nonclean rows clauses designated record) : | |
| HEq (leftChild rows clauses designated record i) | |
| (physicalLeft rows clauses designated i.val (extract rows clauses designated record i)) := | |
| cast_heq _ _ | |
| private theorem dependent_congr {A : Sort*} {B : A → Sort*} (f : (a : A) → B a) | |
| {a b : A} (h : a = b) : HEq (f a) (f b) := by | |
| cases h | |
| rfl | |
| theorem rightChild_event_heq (record : Record (C := C) (t := t) rows clauses designated) | |
| (x : Sample (C := C) (t := t) rows clauses designated) | |
| (hevent : event rows clauses designated record x = true) | |
| (i : Nonclean rows clauses designated record) : | |
| HEq (rightChild rows clauses designated record i) | |
| (SourceChildKernel.rightChildren rows clauses designated | |
| (fun j => ((x j).1, (x j).2.1)) (fun j => (x j).2.2) i.val) := | |
| (rightChild_heq rows clauses designated record i).trans | |
| (dependent_congr (physicalRight rows clauses designated i.val) | |
| (extract_eq rows clauses designated record x hevent i)) | |
| theorem leftChild_event_heq (record : Record (C := C) (t := t) rows clauses designated) | |
| (x : Sample (C := C) (t := t) rows clauses designated) | |
| (hevent : event rows clauses designated record x = true) | |
| (i : Nonclean rows clauses designated record) : | |
| HEq (leftChild rows clauses designated record i) | |
| (SourceChildKernel.leftChildren rows clauses designated | |
| (fun j => ((x j).1, (x j).2.1)) (fun j => (x j).2.2) i.val) := by | |
| have hphysical : | |
| HEq (physicalLeft rows clauses designated i.val (extract rows clauses designated record i)) | |
| (physicalLeft rows clauses designated i.val (x i.val)) := | |
| dependent_congr (physicalLeft rows clauses designated i.val) | |
| (extract_eq rows clauses designated record x hevent i) | |
| exact (leftChild_heq rows clauses designated record i).trans | |
| (hphysical.trans (heq_of_eq (physicalLeft_eq_leftChildren rows clauses designated x i.val))) | |
| omit [NeZero m] in | |
| theorem childClean_of_clean (record : Record (C := C) (t := t) rows clauses designated) | |
| (x : Sample (C := C) (t := t) rows clauses designated) | |
| (hevent : event rows clauses designated record x = true) (i : Fin (branch h)) | |
| (hi : clean rows clauses designated record i) : | |
| UniformCleanSources.childClean | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) | |
| (SourceChildKernel.zero (C := C) (t := t) rows clauses designated) | |
| i (x i).1 (x i).2.1 (x i).2.2 = true := by | |
| have obs := CleanSourceRecord.observation_eq designated | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) | |
| (SourceChildKernel.zero (C := C) (t := t) rows clauses designated) record x hevent i | |
| cases hc : UniformCleanSources.childClean | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) | |
| (SourceChildKernel.zero (C := C) (t := t) rows clauses designated) | |
| i (x i).1 (x i).2.1 (x i).2.2 with | |
| | false => | |
| have hr : record i = .inl (x i) := by | |
| simpa only [SelectedSources.childObservation, hc, Bool.false_eq_true, ite_false] | |
| using obs.symm | |
| simp only [clean, CleanSourceRecord.clean, hr, UniformCleanSources.cleanRecord, | |
| Bool.false_eq_true] at hi | |
| | true => rfl | |
| end | |
| end PerfectCompleteness.CleanRecordChildBlocks | |
| end | |
| end OAI | |