openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Repetition /CleanEndpointSlots.lean
| import OAI.Computability.PerfectCompleteness.Construction.SourceQuestionReconstruction | |
| import OAI.Computability.PerfectCompleteness.Foundations.CutSlotAssembly | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CleanEndpointSlots | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces | |
| open SourceQuestionReconstruction | |
| variable {branch : Nat → Nat} {n h t v m : Nat} [NeZero m] | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch h) → Slots branch h) | |
| (clean : Fin (branch h) → Prop) | |
| (visible : Visible (E := SourceTuple m t) designated clean) | |
| def leftInside (own : {i : Fin (branch h) // clean i} → Fin t → Fin m) : | |
| Slots branch (h + 1) → Fin t → MixedSupport.Slot := | |
| fun s k => SourceKeys.slot clauses | |
| (reconstructLeft designated clean visible own s.1 s.2 k) | |
| def rightInside (projected : Fin (branch h) → Bool) | |
| (own : {i : Fin (branch h) // clean i} → Fin t → Fin v) : | |
| Slots branch (h + 1) → Fin t → MixedSupport.Slot := | |
| fun s k => SourceKeys.slot clauses | |
| (reconstructRight designated clean clauses projected visible own s.1 s.2 k) | |
| omit [NeZero m] in | |
| theorem leftInside_nonclean | |
| (own other : {i : Fin (branch h) // clean i} → Fin t → Fin m) | |
| (i : Fin (branch h)) (hi : ¬ clean i) (s : Slots branch h) : | |
| leftInside clauses designated clean visible own (i, s) = | |
| leftInside clauses designated clean visible other (i, s) := by | |
| funext k | |
| have hn : ¬ (clean i ∧ s = designated i) := fun h => hi h.1 | |
| simp only [leftInside, reconstructLeft, reconstruct, dite_eq_right hn] | |
| omit [NeZero m] in | |
| theorem rightInside_nonclean (projected : Fin (branch h) → Bool) | |
| (own other : {i : Fin (branch h) // clean i} → Fin t → Fin v) | |
| (i : Fin (branch h)) (hi : ¬ clean i) (s : Slots branch h) : | |
| rightInside clauses designated clean visible projected own (i, s) = | |
| rightInside clauses designated clean visible projected other (i, s) := by | |
| funext k | |
| have hn : ¬ (clean i ∧ s = designated i) := fun h => hi h.1 | |
| simp only [rightInside, reconstructRight, reconstruct, dite_eq_right hn] | |
| def leftReference : Slots branch (h + 1) → Fin t → MixedSupport.Slot := | |
| leftInside clauses designated clean visible (fun _ _ => 0) | |
| def rightReference (projected : Fin (branch h) → Bool) : | |
| Slots branch (h + 1) → Fin t → MixedSupport.Slot := | |
| rightInside clauses designated clean visible projected | |
| (fun _ _ => (clauses 0).variable 0) | |
| variable (rows repeats : Nat → Nat) (p : Path branch n (h + 1)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| def leftSlots (own : {i : Fin (branch h) // clean i} → Fin t → Fin m) := | |
| CutSlotAssembly.fill p outside (leftInside clauses designated clean visible own) | |
| def rightSlots (projected : Fin (branch h) → Bool) | |
| (own : {i : Fin (branch h) // clean i} → Fin t → Fin v) := | |
| CutSlotAssembly.fill p outside (rightInside clauses designated clean visible projected own) | |
| omit [NeZero m] in | |
| theorem leftSlots_kept_eq | |
| (own other : {i : Fin (branch h) // clean i} → Fin t → Fin m) | |
| (s : Slots branch n) (hs : CutSamplerKeyLocality.keptLeaf p clean s) : | |
| leftSlots clauses designated clean visible p outside own s = | |
| leftSlots clauses designated clean visible p outside other s := | |
| CutSlotAssembly.fill_kept_eq p outside _ _ clean | |
| (leftInside_nonclean clauses designated clean visible own other) s hs | |
| omit [NeZero m] in | |
| theorem rightSlots_kept_eq (projected : Fin (branch h) → Bool) | |
| (own other : {i : Fin (branch h) // clean i} → Fin t → Fin v) | |
| (s : Slots branch n) (hs : CutSamplerKeyLocality.keptLeaf p clean s) : | |
| rightSlots clauses designated clean visible p outside projected own s = | |
| rightSlots clauses designated clean visible p outside projected other s := | |
| CutSlotAssembly.fill_kept_eq p outside _ _ clean | |
| (rightInside_nonclean clauses designated clean visible projected own other) s hs | |
| def leftReplay | |
| (record : WholeCutReplay.Tape rows repeats p | |
| (CutSlotAssembly.fill p outside (leftReference clauses designated clean visible)) clean) | |
| (own : {i : Fin (branch h) // clean i} → Fin t → Fin m) : | |
| WholeCutReplay.Tape rows repeats p | |
| (leftSlots clauses designated clean visible p outside own) clean := | |
| CutSlotAssembly.transport rows repeats p outside | |
| (leftReference clauses designated clean visible) | |
| (leftInside clauses designated clean visible own) clean | |
| (leftInside_nonclean clauses designated clean visible (fun _ _ => 0) own) record | |
| def rightReplay (projected : Fin (branch h) → Bool) | |
| (record : WholeCutReplay.Tape rows repeats p | |
| (CutSlotAssembly.fill p outside | |
| (rightReference clauses designated clean visible projected)) clean) | |
| (own : {i : Fin (branch h) // clean i} → Fin t → Fin v) : | |
| WholeCutReplay.Tape rows repeats p | |
| (rightSlots clauses designated clean visible p outside projected own) clean := | |
| CutSlotAssembly.transport rows repeats p outside | |
| (rightReference clauses designated clean visible projected) | |
| (rightInside clauses designated clean visible projected own) clean | |
| (rightInside_nonclean clauses designated clean visible projected | |
| (fun _ _ => (clauses 0).variable 0) own) record | |
| end | |
| end PerfectCompleteness.CleanEndpointSlots | |
| end | |
| end OAI | |