openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Construction /NativeLeafAlphabet.lean
| import OAI.Computability.PerfectCompleteness.Foundations.SourceOddListsLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.NativeLeafAlphabet | |
| open SourceClause | |
| variable {I : Type*} {v m t : Nat} | |
| def castAssignment {a b : I → MixedSupport.Slot} (h : a = b) : | |
| MixedSupport.Assignment a ≃ MixedSupport.Assignment b := | |
| Equiv.cast (congrArg (fun slots : I → MixedSupport.Slot => | |
| MixedSupport.Assignment slots) h) | |
| @[simp] theorem castAssignment_rfl (a : I → MixedSupport.Slot) : | |
| castAssignment (rfl : a = a) = Equiv.refl (MixedSupport.Assignment a) := rfl | |
| def nativeLeftEquiv (clauses : Fin m → NormalizedClause v) | |
| (question : Fin t → Fin m) {a : Fin t → MixedSupport.Slot} | |
| (ha : a = SourceAnswerEquiv.leftSlots clauses question) : | |
| MixedSupport.Assignment a ≃ SourceOddLists.LeftLabels t := | |
| (castAssignment ha).trans (SourceAnswerEquiv.leftEquiv clauses question) | |
| def nativeRightEquiv (clauses : Fin m → NormalizedClause v) | |
| (question : Fin t → Fin v) {b : Fin t → MixedSupport.Slot} | |
| (hb : b = SourceAnswerEquiv.rightSlots clauses question) : | |
| MixedSupport.Assignment b ≃ SourceOddLists.RightLabels t := | |
| (castAssignment hb).trans (SourceAnswerEquiv.rightEquiv clauses question) | |
| theorem projection_encode (clauses : Fin m → NormalizedClause v) | |
| (e : SourceOddLists.Occurrences m t) {a b : Fin t → MixedSupport.Slot} | |
| (ha : a = SourceAnswerEquiv.leftSlots clauses (SourceOddLists.left e)) | |
| (hb : b = SourceAnswerEquiv.rightSlots clauses (SourceOddLists.right clauses e)) | |
| (q : ∀ k, MixedSupport.Projection (a k) (b k)) | |
| (hq : ∀ k, HEq (q k) (SourceAnswerEquiv.slotProjection clauses e k)) | |
| (x : MixedSupport.Assignment a) : | |
| nativeRightEquiv clauses (SourceOddLists.right clauses e) hb | |
| (MixedSupport.projectionMap q x) = | |
| SourceOddLists.projection clauses e | |
| (nativeLeftEquiv clauses (SourceOddLists.left e) ha x) := by | |
| subst a | |
| subst b | |
| have hprojection : q = SourceAnswerEquiv.slotProjection clauses e := by | |
| funext k | |
| exact eq_of_heq (hq k) | |
| subst q | |
| exact SourceAnswerEquiv.projection_encode clauses e x | |
| end PerfectCompleteness.NativeLeafAlphabet | |
| end | |
| end OAI | |