openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /CleanRecordProjection.lean
| import OAI.Computability.PerfectCompleteness.Decoding.CleanDecoderOddLists | |
| import OAI.Computability.PerfectCompleteness.Decoding.CleanEndpointProjection | |
| import OAI.Computability.PerfectCompleteness.Machines.CleanRecordDecoderInput | |
| import OAI.Computability.PerfectCompleteness.Repetition.CleanRecordObservation | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CleanRecordProjection | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces | |
| variable {branch : Nat → Nat} {h t v m : Nat} [NeZero m] | |
| {C : Type*} [Fintype C] | |
| (rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch h) → Slots branch h) | |
| omit [NeZero m] in | |
| private theorem sourceProjection_congr (b b' : Bool) | |
| (e e' : SourceOddLists.Occurrences m t) (hb : b = b') (he : e = e') (k : Fin t) : | |
| HEq (CleanEndpointProjection.sourceProjection clauses b e k) | |
| (CleanEndpointProjection.sourceProjection clauses b' e' k) := by | |
| cases hb | |
| cases he | |
| rfl | |
| omit [NeZero m] in | |
| private theorem sourceProjection_tree (i : Fin (branch h)) | |
| (source : SourceChildKernel.Source (m := m) (t := t) designated i) | |
| (b : Bool) (leaf : Slots branch h) (k : Fin t) : | |
| HEq (CleanEndpointProjection.sourceProjection clauses b | |
| (SourceChildKernel.tree designated i source leaf) k) | |
| (SourceChildKernel.projection clauses designated i source b leaf k) := by | |
| cases b <;> rfl | |
| omit [NeZero m] in | |
| theorem insideProjection_heq | |
| (record : CleanRecordObservation.Record (C := C) (t := t) rows clauses designated) | |
| (x : CleanRecordObservation.Sample (C := C) (t := t) rows clauses designated) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (s : Slots branch (h + 1)) (k : Fin t) : | |
| HEq | |
| (CleanEndpointProjection.insideProjection clauses designated | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (CleanRecordChildBlocks.visible rows clauses designated record) | |
| (CleanRecordChildBlocks.projected rows clauses designated record) | |
| (CleanSourceRecord.clean_projected designated | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) record) | |
| (fun i => (x i.val).1) s k) | |
| (SourceChildKernel.parentProjection rows clauses designated | |
| (fun i => ((x i).1, (x i).2.1)) (fun i => (x i).2.2) s k) := by | |
| have hp := congrFun (CleanSourceRecord.projected_eq_raw designated | |
| (D := SourceChildKernel.D (C := C) (t := t) rows clauses designated) | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) | |
| (SourceChildKernel.zero (C := C) (t := t) rows clauses designated) | |
| record x hevent) s.1 | |
| have hflag : CleanSourceRecord.rawProjected designated | |
| (D := SourceChildKernel.D (C := C) (t := t) rows clauses designated) | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) | |
| s.1 (x s.1).2.2 = | |
| SourceChildKernel.rawProjected rows clauses designated s.1 (x s.1).2.2 := by | |
| cases (x s.1).2.2 <;> rfl | |
| by_cases hs : CleanRecordChildBlocks.clean rows clauses designated record s.1 ∧ | |
| s.2 = designated s.1 | |
| · simp only [CleanEndpointProjection.insideProjection, dite_eq_left hs] | |
| apply (CutProjectionAssembly.castProjection_heq _ _ _).trans | |
| have he : (x s.1).1 = SourceChildKernel.tree designated s.1 | |
| ((x s.1).1, (x s.1).2.1) s.2 := by | |
| rw [hs.2] | |
| exact (CleanSourceRecord.assemble_designated designated s.1 _ _).symm | |
| exact (sourceProjection_congr clauses _ _ _ _ (hp.trans hflag) he k).trans | |
| (sourceProjection_tree clauses designated s.1 ((x s.1).1, (x s.1).2.1) | |
| (SourceChildKernel.rawProjected rows clauses designated s.1 (x s.1).2.2) s.2 k) | |
| · simp only [CleanEndpointProjection.insideProjection, dite_eq_right hs] | |
| apply (CutProjectionAssembly.castProjection_heq _ _ _).trans | |
| have hv := CleanSourceRecord.visible_eq_reveal designated | |
| (D := SourceChildKernel.D (C := C) (t := t) rows clauses designated) | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) | |
| (SourceChildKernel.zero (C := C) (t := t) rows clauses designated) | |
| record x hevent | |
| have he : CleanRecordChildBlocks.visible rows clauses designated record s.1 ⟨s.2, hs⟩ = | |
| SourceChildKernel.tree designated s.1 ((x s.1).1, (x s.1).2.1) s.2 := | |
| congrArg (fun visible => visible s.1 ⟨s.2, hs⟩) hv | |
| exact (sourceProjection_congr clauses _ _ _ _ (hp.trans hflag) he k).trans | |
| (sourceProjection_tree clauses designated s.1 ((x s.1).1, (x s.1).2.1) | |
| (SourceChildKernel.rawProjected rows clauses designated s.1 (x s.1).2.2) s.2 k) | |
| private theorem fillProjection_congr {n : Nat} (p : Path branch n (h + 1)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| {left left' right right' : Slots branch (h + 1) → Fin t → MixedSupport.Slot} | |
| (hl : left = left') (hr : right = right') | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) | |
| (q' : ∀ s k, MixedSupport.Projection (left' s k) (right' s k)) | |
| (hq : ∀ s k, HEq (q s k) (q' s k)) (s : Slots branch n) (k : Fin t) : | |
| HEq (CutProjectionAssembly.fillProjection p outside left right q s k) | |
| (CutProjectionAssembly.fillProjection p outside left' right' q' s k) := by | |
| cases hl | |
| cases hr | |
| have hqq : q = q' := funext (fun s => funext (fun k => eq_of_heq (hq s k))) | |
| cases hqq | |
| rfl | |
| omit [NeZero m] in | |
| theorem fillProjection_heq {n : Nat} (p : Path branch n (h + 1)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (record : CleanRecordObservation.Record (C := C) (t := t) rows clauses designated) | |
| (x : CleanRecordObservation.Sample (C := C) (t := t) rows clauses designated) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (s : Slots branch n) (k : Fin t) : | |
| HEq | |
| (CutProjectionAssembly.fillProjection p outside _ _ | |
| (CleanEndpointProjection.insideProjection clauses designated | |
| (CleanRecordChildBlocks.clean rows clauses designated record) | |
| (CleanRecordChildBlocks.visible rows clauses designated record) | |
| (CleanRecordChildBlocks.projected rows clauses designated record) | |
| (CleanSourceRecord.clean_projected designated | |
| (SourceChildKernel.Factors (C := C) (t := t) rows clauses designated) record) | |
| (fun i => (x i.val).1)) s k) | |
| (CutProjectionAssembly.fillProjection p outside _ _ | |
| (SourceChildKernel.parentProjection rows clauses designated | |
| (fun i => ((x i).1, (x i).2.1)) (fun i => (x i).2.2)) s k) := | |
| fillProjection_congr p outside | |
| (CleanRecordObservation.leftInside_eq rows clauses designated record x hevent) | |
| (CleanRecordObservation.rightInside_eq rows clauses designated record x hevent) _ _ | |
| (insideProjection_heq rows clauses designated record x hevent) s k | |
| theorem decoderProjection_heq {root : Nat} (repeats : Nat → Nat) | |
| (upper : HierarchicalArrays.Nodes branch root) | |
| (d : HierarchicalFrozenTables.LowerNodes upper (h + 1)) | |
| (cut : OwnInputReference.Cut upper (HierarchicalLeftDecoder.LowerNode upper (h + 1) d)) | |
| (outside : Slots branch root → Fin t → MixedSupport.Slot) | |
| (placeholder : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (record : CleanRecordDecoderInput.Record (t := t) rows repeats upper d cut clauses designated) | |
| (external : CleanRecordDecoderInput.Exterior rows repeats upper d cut outside placeholder) | |
| (x : CleanRecordDecoderInput.Sample (t := t) rows repeats upper d cut clauses designated) | |
| (hevent : CleanRecordChildBlocks.event rows clauses designated record x = true) | |
| (s : Slots branch root) (k : Fin t) : | |
| HEq (CleanDecoderOddLists.fullProjection | |
| (CleanRecordDecoderInput.exposed rows repeats upper d cut outside placeholder | |
| clauses designated record external) | |
| (CleanRecordDecoderInput.sources rows repeats upper d cut clauses designated record x) s k) | |
| (CutProjectionAssembly.fillProjection (CleanRecordDecoderInput.path upper d cut) outside _ _ | |
| (SourceChildKernel.parentProjection rows clauses designated | |
| (fun i => ((x i).1, (x i).2.1)) (fun i => (x i).2.2)) s k) := | |
| fillProjection_heq rows clauses designated (CleanRecordDecoderInput.path upper d cut) | |
| outside record x hevent s k | |
| end | |
| end PerfectCompleteness.CleanRecordProjection | |
| end | |
| end OAI | |