openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /DomainSyntaxClosure.lean
| import OAI.Computability.DegreeRigidity.Constructibility.DomainParameterizedSyntax | |
| namespace OAI | |
| namespace TuringRigidity.BoundedDefinability | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SetModelFunctions | |
| universe u | |
| variable {M : ZFSet.{u}} {P Q : ZFSet.{u} → (ℕ → ZFSet.{u}) → Prop} | |
| theorem DomainDefinable.neg (h : DomainDefinable M P) : | |
| DomainDefinable M (fun N e => ¬ P N e) := by | |
| obtain ⟨p,d,k,hk,hd,hp⟩ := h | |
| exact ⟨.neg p,d,k,hk,hd,fun N hN hdN e he => not_congr (hp N hN hdN e he)⟩ | |
| theorem DomainDefinable.imp (hP : DomainDefinable M P) (hQ : DomainDefinable M Q) : | |
| DomainDefinable M (fun N e => P N e → Q N e) := by | |
| apply (hP.and hQ.neg).neg.congr | |
| intro N e | |
| classical | |
| tauto | |
| theorem DomainDefinable.subst (h : DomainDefinable M P) (r : ℕ → ℕ) : | |
| DomainDefinable M (fun N e => P N (e ∘ r)) := by | |
| obtain ⟨p,d,k,hk,hd,hp⟩ := h | |
| refine ⟨p.rename (slotMap r id),d,k,hk,hd,?_⟩ | |
| intro N hN hdN e he | |
| rw [SentenceForm.sat_rename] | |
| change p.Sat _ (mix e d ∘ slotMap r id) ↔ _ | |
| rw [mix_slotMap,Function.comp_id] | |
| exact hp N hN hdN (e ∘ r) (fun i => he (r i)) | |
| theorem DomainDefinable.allMem (C : Context M) (h : DomainDefinable M P) (a : ℕ) : | |
| DomainDefinable M (fun N e => ∀ x ∈ e a, P N (cons x e)) := by | |
| obtain ⟨p,d,k,hk,hd,hp⟩ := ((member_definable C 0 (a+1)).toDomain.and h.neg).existsSet.neg | |
| refine ⟨p,d,k,hk,hd,?_⟩ | |
| intro N hN hdN e he | |
| rw [hp N hN hdN e he] | |
| change (¬ ∃ x ∈ N, x ∈ e a ∧ ¬ P N (cons x e)) ↔ _ | |
| classical | |
| constructor | |
| · intro h x hx | |
| by_contra hn | |
| exact h ⟨x,hN _ (he a) x hx,hx,hn⟩ | |
| · intro h ⟨x,_,hx,hn⟩ | |
| exact hn (h x hx) | |
| end TuringRigidity.BoundedDefinability | |
| end OAI | |