| 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 |
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|
