File size: 3,514 Bytes
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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 | import OAI.Computability.DegreeRigidity.OrdinalCodes.NumericDefSystem
import OAI.Computability.DegreeRigidity.Constructibility.DomainSyntaxClosure
namespace OAI
namespace TuringRigidity.NumericSyntax
open ElementaryModel BoundedSetTheory TransitiveNameModel BoundedDefinability
universe u
def Domain (P : ZFSet.{u} → (ℕ → ZFSet.{u}) → Prop) : Prop :=
∃ p : SentenceForm, ∀ B, Covers B → ∀ N : ZFSet.{u}, Transitive N →
(∀ i, params B i ∈ N) → ∀ e, (∀ i, e i ∈ N) →
(p.Sat (N : Set ZFSet) (mix e (params B)) ↔ P N e)
variable {P Q : ZFSet.{u} → (ℕ → ZFSet.{u}) → Prop}
theorem Numeric.toDomain {P : (ℕ → ZFSet.{u}) → Prop} (h : Numeric P) :
Domain (fun _ e => P e) := by
obtain ⟨p,hp⟩ := h
refine ⟨fromBounded p,?_⟩
intro B hB N hN hd e he
rw [bounded_sat,Formula.absolute p N hN _ (by
intro i; unfold mix; split <;> first | exact he _ | exact hd _)]
exact hp B hB e
theorem Domain.congr (h : Domain P) (he : ∀ N e, P N e ↔ Q N e) : Domain Q := by
obtain ⟨p,hp⟩ := h
exact ⟨p,fun B hB N hN hd e he' => (hp B hB N hN hd e he').trans (he N e)⟩
theorem Domain.and (h : Domain P) (k : Domain Q) : Domain (fun N e => P N e ∧ Q N e) := by
obtain ⟨p,hp⟩ := h; obtain ⟨q,hq⟩ := k
exact ⟨.conj p q,fun B hB N hN hd e he =>
and_congr (hp B hB N hN hd e he) (hq B hB N hN hd e he)⟩
theorem Domain.neg (h : Domain P) : Domain (fun N e => ¬ P N e) := by
obtain ⟨p,hp⟩ := h
exact ⟨.neg p,fun B hB N hN hd e he => not_congr (hp B hB N hN hd e he)⟩
theorem Domain.imp (h : Domain P) (k : Domain Q) : Domain (fun N e => P N e → Q N e) :=
(h.and k.neg).neg.congr (fun _ _ => by tauto)
theorem Domain.subst (h : Domain P) (r : ℕ → ℕ) : Domain (fun N e => P N (e ∘ r)) := by
obtain ⟨p,hp⟩ := h
refine ⟨p.rename (slotMap r id),?_⟩
intro B hB N hN hd e he
rw [SentenceForm.sat_rename]
change p.Sat _ (mix e (params B) ∘ slotMap r id) ↔ _
rw [mix_slotMap,Function.comp_id]
exact hp B hB N hN hd (e ∘ r) (fun i => he (r i))
theorem Domain.existsSet (h : Domain P) : Domain (fun N e => ∃ x ∈ N, P N (cons x e)) := by
obtain ⟨p,hp⟩ := h
refine ⟨.ex (p.rename bindSlots),?_⟩
intro B hB N hN hd e he
change (∃ x ∈ N, (p.rename bindSlots).Sat _ (cons x (mix e (params B)))) ↔ _
apply exists_congr; intro x
apply and_congr_right; intro hx
rw [SentenceForm.sat_rename]
change p.Sat _ (cons x (mix e (params B)) ∘ bindSlots) ↔ _
rw [bind_mix]
exact hp B hB N hN hd (cons x e) (by intro i; cases i; exact hx; exact he _)
theorem Domain.allMem (h : Domain P) (a : ℕ) : Domain (fun N e => ∀ x ∈ e a, P N (cons x e)) := by
obtain ⟨p,hp⟩ := ((member_definable 0 (a+1)).toDomain.and h.neg).existsSet.neg
refine ⟨p,?_⟩
intro B hB N hN hd e he
rw [hp B hB N hN hd 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)
theorem powerBound_domain (a q : ℕ) : Domain (fun N e => RelativeConstructible.PowerBound N (e a) (e q)) := by
refine ⟨FullSetForcing.ownPower (2*a) (2*q),?_⟩
intro B hB N hN hd e he
rw [FullSetForcing.ownPower_spec N hN _ _ _ (by
intro i; unfold mix; split <;> first | exact he _ | exact hd _)]
simp only [mix_even,RelativeConstructible.PowerBound]
end TuringRigidity.NumericSyntax
end OAI
|