| import OAI.Computability.DegreeRigidity.Syntax.LevySigmaNormalization |
|
|
| namespace OAI |
|
|
|
|
| namespace TuringRigidity.BoundedSetTheory |
| open TransitiveNameModel |
| universe u |
|
|
| theorem levy_sigma_schemas (M : ZFSet.{u}) (hM : Transitive M) |
| (hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) |
| (hS : SigmaSeparation M) (hR : SigmaReplacement M) (hI : Infinity M) (hAC : Choice M) : |
| LevySigmaSeparation M ∧ LevySigmaReplacement M := by |
| have hC := sigma_collection M hM hP hU hPow hS hR hI hAC |
| have correct := LevySigma.normalize_correct M hM hP hU hS.bounded hR hI hC |
| constructor |
| · intro φ e he a ha |
| obtain ⟨b,hb,hbdef⟩ := hS φ.normalize e he a ha |
| refine ⟨b,hb,?_⟩ |
| intro x hx |
| rw [hbdef x hx,correct φ (cons x e) (by intro i; cases i <;> simp [cons,he,hx])] |
| · intro φ e he a ha htotal |
| have eq (x y : ZFSet.{u}) (hx : x ∈ M) (hy : y ∈ M) : |
| φ.normalize.Realize M (cons y (cons x e)) ↔ φ.Realize M (cons y (cons x e)) := |
| correct φ _ (by |
| intro i; rcases i with _|i; exact hy |
| rcases i with _|i; exact hx |
| exact he i) |
| obtain ⟨b,hb,hbdef⟩ := hR φ.normalize e he a ha (by |
| intro x hx |
| have hxM := hM a ha x hx |
| obtain ⟨y,hy,hφ,hu⟩ := htotal x hx |
| exact ⟨y,hy,(eq x y hxM hy).mpr hφ,fun z hz h => hu z hz ((eq x z hxM hz).mp h)⟩) |
| refine ⟨b,hb,?_⟩ |
| intro y hy |
| rw [hbdef y hy] |
| apply exists_congr; intro x |
| exact and_congr_right (fun hx => eq x y (hM a ha x hx) hy) |
|
|
| def SigmaFormula.toLevy : SigmaFormula → LevySigma |
| | .bounded φ => .bounded φ |
| | .existsSet φ => .existsSet φ.toLevy |
|
|
| theorem SigmaFormula.realize_toLevy (φ : SigmaFormula) (M : ZFSet.{u}) (e : ℕ → ZFSet.{u}) : |
| φ.toLevy.Realize M e ↔ φ.Realize M e := by |
| induction φ generalizing e with |
| | bounded φ => rfl |
| | existsSet φ ih => |
| simp only [toLevy,LevySigma.Realize,Realize] |
| apply exists_congr; intro x |
| exact and_congr_right (fun _ => ih (cons x e)) |
|
|
| theorem LevySigmaSeparation.finitePrefix {M : ZFSet.{u}} (hS : LevySigmaSeparation M) : SigmaSeparation M := by |
| intro φ e he a ha |
| obtain ⟨b,hb,hbdef⟩ := hS φ.toLevy e he a ha |
| exact ⟨b,hb,fun x hx => (hbdef x hx).trans (and_congr_right (fun _ => φ.realize_toLevy M _))⟩ |
|
|
| theorem LevySigmaReplacement.finitePrefix {M : ZFSet.{u}} (hR : LevySigmaReplacement M) : SigmaReplacement M := by |
| intro φ e he a ha htotal |
| obtain ⟨b,hb,hbdef⟩ := hR φ.toLevy e he a ha (by |
| intro x hx |
| obtain ⟨y,hy,hφ,hu⟩ := htotal x hx |
| exact ⟨y,hy,(φ.realize_toLevy M _).mpr hφ,fun z hz h => hu z hz ((φ.realize_toLevy M _).mp h)⟩) |
| refine ⟨b,hb,?_⟩ |
| intro y hy |
| rw [hbdef y hy] |
| apply exists_congr; intro x |
| exact and_congr_right (fun _ => φ.realize_toLevy M _) |
|
|
| end TuringRigidity.BoundedSetTheory |
|
|
| end OAI |
|
|
