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9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 | 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
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