File size: 2,850 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 | 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
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