File size: 2,664 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 | import OAI.Computability.DegreeRigidity.Syntax.SigmaSyntax
namespace OAI
namespace TuringRigidity.BoundedSetTheory
universe u
theorem map_cons (j : ZFSet.{u} → ZFSet.{u}) (x : ZFSet.{u}) (e : ℕ → ZFSet.{u}) :
(fun i => j (cons x e i)) = cons (j x) (fun i => j (e i)) := by
funext i; cases i <;> rfl
theorem Formula.realize_isomorphism (A B : ZFSet.{u}) (j : ZFSet.{u} → ZFSet.{u})
(hin : ∀ x ∈ A, j x ∈ B)
(hout : ∀ y ∈ B, ∃ x ∈ A, j x = y)
(hi : ∀ x ∈ A, ∀ y ∈ A, j x = j y → x = y)
(hm : ∀ x ∈ A, ∀ y ∈ A, j x ∈ j y ↔ x ∈ y)
(φ : Formula) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ A) :
φ.Realize A e ↔ φ.Realize B (fun i => j (e i)) := by
induction φ generalizing e with
| equal i k =>
exact ⟨congrArg j,hi _ (he i) _ (he k)⟩
| member i k => exact (hm _ (he i) _ (he k)).symm
| conj φ ψ ihφ ihψ => exact and_congr (ihφ e he) (ihψ e he)
| neg φ ih => exact not_congr (ih e he)
| existsMem i φ ih =>
have he' (x : ZFSet.{u}) (hx : x ∈ A) : ∀ n, cons x e n ∈ A := by
intro n; cases n with
| zero => exact hx
| succ n => exact he n
constructor
· rintro ⟨x,hx,hxi,hφ⟩
refine ⟨j x,hin x hx,(hm x hx (e i) (he i)).mpr hxi,?_⟩
have h := (ih (cons x e) (he' x hx)).mp hφ
rwa [map_cons] at h
· rintro ⟨y,hy,hyi,hφ⟩
obtain ⟨x,hx,rfl⟩ := hout y hy
refine ⟨x,hx,(hm x hx (e i) (he i)).mp hyi,?_⟩
apply (ih (cons x e) (he' x hx)).mpr
rwa [map_cons]
theorem SigmaFormula.realize_isomorphism (A B : ZFSet.{u}) (j : ZFSet.{u} → ZFSet.{u})
(hin : ∀ x ∈ A, j x ∈ B)
(hout : ∀ y ∈ B, ∃ x ∈ A, j x = y)
(hi : ∀ x ∈ A, ∀ y ∈ A, j x = j y → x = y)
(hm : ∀ x ∈ A, ∀ y ∈ A, j x ∈ j y ↔ x ∈ y)
(φ : SigmaFormula) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ A) :
φ.Realize A e ↔ φ.Realize B (fun i => j (e i)) := by
induction φ generalizing e with
| bounded φ => exact φ.realize_isomorphism A B j hin hout hi hm e he
| existsSet φ ih =>
have he' (x : ZFSet.{u}) (hx : x ∈ A) : ∀ n, cons x e n ∈ A := by
intro n; cases n with
| zero => exact hx
| succ n => exact he n
constructor
· rintro ⟨x,hx,hφ⟩
refine ⟨j x,hin x hx,?_⟩
have h := (ih (cons x e) (he' x hx)).mp hφ
rwa [map_cons] at h
· rintro ⟨y,hy,hφ⟩
obtain ⟨x,hx,rfl⟩ := hout y hy
apply Exists.intro x
refine ⟨hx,?_⟩
apply (ih (cons x e) (he' x hx)).mpr
rwa [map_cons]
end TuringRigidity.BoundedSetTheory
end OAI
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