| import Mathlib |
| import OAI.Computability.MinUncut.Graphs.SignedGraph |
| import OAI.Computability.MinUncut.Analysis.GaussianRows |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace MinUncut.Outer |
| open MinUncut.Inner MinUncut.FiniteProof |
| attribute [local instance] Classical.propDecidable |
| variable {Name I : Type*} [Fintype I] |
|
|
| abbrev FamilyQuestion (Name I : Type*) := (I → Equation Name) ⊕ (I → SecondQuestion Name) |
|
|
| def FamilyAlphabet : FamilyQuestion Name I → Type _ |
| | .inl U => FirstAlphabet U |
| | .inr Q => SecondAlphabet Q |
|
|
| instance familyAlphabetFintype (q : FamilyQuestion Name I) : Fintype (FamilyAlphabet q) := by |
| cases q <;> unfold FamilyAlphabet <;> infer_instance |
|
|
| def familyBase : (q : FamilyQuestion Name I) → FamilyAlphabet q |
| | .inl U => fun j => origin (U j) |
| | .inr _ => fun _ _ => 0 |
|
|
| abbrev FamilyVariable (Name I : Type*) [Fintype I] := Variable (familyBase (Name := Name) (I := I)) |
|
|
| def variableProof (y : FamilyVariable Name I → Bool) : ProofFamily Name I where |
| first U := ⟨answer familyBase y (.inl U),answer_folded familyBase y (.inl U)⟩ |
| second Q := ⟨answer familyBase y (.inr Q),answer_folded familyBase y (.inr Q)⟩ |
|
|
| def ProofFamily.queryAnswer (f : ProofFamily Name I) : |
| (q : FamilyQuestion Name I) → (FamilyAlphabet q → Bool) → Bool |
| | .inl U => (f.first U).answer |
| | .inr Q => (f.second Q).answer |
|
|
| lemma ProofFamily.queryAnswer_folded (f : ProofFamily Name I) : Folded f.queryAnswer := by |
| intro q P |
| cases q with |
| | inl U => exact (f.first U).folded P |
| | inr Q => exact (f.second Q).folded P |
|
|
| def ProofFamily.assignment (f : ProofFamily Name I) : FamilyVariable Name I → Bool := |
| fun p => f.queryAnswer p.1 p.2.val |
|
|
| lemma ProofFamily.answer_assignment (f : ProofFamily Name I) : |
| answer familyBase f.assignment=f.queryAnswer := |
| answer_of_folded familyBase f.queryAnswer f.queryAnswer_folded |
|
|
| lemma foldedProof_ext {A : Type*} {f g : FoldedProof A} (h : f.answer=g.answer) : f=g := by |
| cases f |
| cases g |
| cases h |
| rfl |
|
|
| lemma proofFamily_ext {f g : ProofFamily Name I} |
| (hf : f.first=g.first) (hg : f.second=g.second) : f=g := by |
| cases f |
| cases g |
| cases hf |
| cases hg |
| rfl |
|
|
| lemma ProofFamily.variableProof_assignment (f : ProofFamily Name I) : variableProof f.assignment=f := by |
| apply proofFamily_ext |
| · funext U |
| apply foldedProof_ext |
| exact (congrFun f.answer_assignment (.inl U)) |
| · funext Q |
| apply foldedProof_ext |
| exact (congrFun f.answer_assignment (.inr Q)) |
|
|
| lemma variableProof_queryAnswer (y : FamilyVariable Name I → Bool) : |
| (variableProof y).queryAnswer=answer familyBase y := by |
| funext q |
| cases q <;> rfl |
| end MinUncut.Outer |
|
|
| end |
| end |
|
|
| end OAI |
|
|
