File size: 3,560 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 88 89 90 91 92 | import OAI.Computability.DegreeRigidity.SetModels.ModelProducts
namespace OAI
namespace TuringRigidity.BoundedSetTheory
open TransitiveNameModel
universe u
namespace Formula
def imp (φ ψ : Formula) : Formula := disj (.neg φ) ψ
def iff (φ ψ : Formula) : Formula := .conj (imp φ ψ) (imp ψ φ)
@[simp] theorem eval_imp (φ ψ : Formula) (e : ℕ → ZFSet.{u}) :
(imp φ ψ).Eval e ↔ (φ.Eval e → ψ.Eval e) := by
classical
simp only [imp, eval_disj, Eval]
constructor
· intro h hp; exact h.elim (fun hn => False.elim (hn hp)) id
· intro h; by_cases hp : φ.Eval e
· exact Or.inr (h hp)
· exact Or.inl hp
@[simp] theorem eval_iff (φ ψ : Formula) (e : ℕ → ZFSet.{u}) :
(iff φ ψ).Eval e ↔ (φ.Eval e ↔ ψ.Eval e) := by
simp only [iff, Eval, eval_imp, iff_def]
def subset (a b : ℕ) : Formula := allMem a (.member 0 (b+1))
def transitive (a : ℕ) : Formula := allMem a (subset 0 (a+1))
@[simp] theorem eval_subset (a b : ℕ) (e : ℕ → ZFSet.{u}) :
(subset a b).Eval e ↔ e a ⊆ e b := by
simp only [subset, eval_allMem, Eval, cons_zero, cons_succ]
rfl
@[simp] theorem eval_transitive (a : ℕ) (e : ℕ → ZFSet.{u}) :
(transitive a).Eval e ↔ Transitive (e a) := by
simp only [transitive, eval_allMem, eval_subset, cons_zero, cons_succ]
rfl
def functionGraph (f d v : ℕ) : Formula :=
.conj (allMem f (.existsMem (d+1) (.existsMem (v+2) (orderedPair 2 1 0))))
(allMem d (.existsMem (v+1)
(.conj (pairMem 1 0 (f+2))
(allMem (v+2) (imp (pairMem 2 0 (f+3)) (.equal 0 1))))))
@[simp] theorem eval_functionGraph (f d v : ℕ) (e : ℕ → ZFSet.{u}) :
(functionGraph f d v).Eval e ↔
(∀ z ∈ e f, ∃ x ∈ e d, ∃ y ∈ e v, z = ZFSet.pair x y) ∧
(∀ x ∈ e d, ∃ y ∈ e v, ZFSet.pair x y ∈ e f ∧
∀ z ∈ e v, ZFSet.pair x z ∈ e f → z = y) := by
simp only [functionGraph, Eval, eval_allMem, eval_orderedPair,
eval_pairMem, eval_imp, cons_zero, cons_succ]
end Formula
inductive SigmaFormula where
| bounded (φ : Formula)
| existsSet (φ : SigmaFormula)
def SigmaFormula.Realize (M : ZFSet.{u}) (e : ℕ → ZFSet.{u}) : SigmaFormula → Prop
| .bounded φ => φ.Realize M e
| .existsSet φ => ∃ x ∈ M, φ.Realize M (cons x e)
def SigmaReplacement (M : ZFSet.{u}) : Prop :=
∀ (φ : SigmaFormula) (e : ℕ → ZFSet.{u}), (∀ i, e i ∈ M) →
∀ a ∈ M,
(∀ x ∈ a, ∃ y ∈ M, φ.Realize M (cons y (cons x e)) ∧
∀ z ∈ M, φ.Realize M (cons z (cons x e)) → z = y) →
∃ b ∈ M, ∀ y ∈ M, y ∈ b ↔ ∃ x ∈ a, φ.Realize M (cons y (cons x e))
theorem replacement_image (M : ZFSet.{u}) (hM : Transitive M)
(hR : SigmaReplacement M) (φ : SigmaFormula) (e : ℕ → ZFSet.{u})
(he : ∀ i, e i ∈ M) {a : ZFSet.{u}} (ha : a ∈ M)
(F : ZFSet.{u} → ZFSet.{u}) (hF : ∀ x ∈ a, F x ∈ M)
(hφ : ∀ x ∈ a, ∀ y ∈ M, φ.Realize M (cons y (cons x e)) ↔ y = F x) :
∃ b ∈ M, ∀ y, y ∈ b ↔ ∃ x ∈ a, F x = y := by
obtain ⟨b,hb,hbdef⟩ := hR φ e he a ha (fun x hx =>
⟨F x,hF x hx,(hφ x hx _ (hF x hx)).mpr rfl,
fun z hz h => (hφ x hx z hz).mp h⟩)
refine ⟨b,hb,fun y => ?_⟩
constructor
· intro hy
have hyM := hM b hb y hy
obtain ⟨x,hx,hxy⟩ := (hbdef y hyM).mp hy
exact ⟨x,hx,((hφ x hx y hyM).mp hxy).symm⟩
· rintro ⟨x,hx,rfl⟩
exact (hbdef _ (hF x hx)).mpr ⟨x,hx,(hφ x hx _ (hF x hx)).mpr rfl⟩
end TuringRigidity.BoundedSetTheory
end OAI
|