File size: 3,418 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 | import OAI.Computability.DegreeRigidity.Syntax.SigmaDisjunction
namespace OAI
namespace TuringRigidity.BoundedSetTheory
open TransitiveNameModel
universe u
inductive LevySigma where
| bounded (φ : Formula)
| conj (φ ψ : LevySigma)
| disj (φ ψ : LevySigma)
| existsSet (φ : LevySigma)
| existsMem (a : ℕ) (φ : LevySigma)
| allMem (a : ℕ) (φ : LevySigma)
namespace LevySigma
def Realize (M : ZFSet.{u}) (e : ℕ → ZFSet.{u}) : LevySigma → Prop
| .bounded φ => φ.Realize M e
| .conj φ ψ => φ.Realize M e ∧ ψ.Realize M e
| .disj φ ψ => φ.Realize M e ∨ ψ.Realize M e
| .existsSet φ => ∃ x ∈ M, φ.Realize M (cons x e)
| .existsMem a φ => ∃ x ∈ M, x ∈ e a ∧ φ.Realize M (cons x e)
| .allMem a φ => ∀ x ∈ M, x ∈ e a → φ.Realize M (cons x e)
def normalize : LevySigma → SigmaFormula
| .bounded φ => .bounded φ
| .conj φ ψ => .conj φ.normalize ψ.normalize
| .disj φ ψ => .disj φ.normalize ψ.normalize
| .existsSet φ => .existsSet φ.normalize
| .existsMem a φ => .existsMem a φ.normalize
| .allMem a φ => .allMem a φ.normalize
theorem normalize_correct (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hS : Separation M)
(hR : SigmaReplacement M) (hI : Infinity M) (hC : SigmaCollection M)
(φ : LevySigma) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) :
φ.normalize.Realize M e ↔ φ.Realize M e := by
have hω := omega_mem M hM hS hI
have h0 := hM _ hω _ ZFSet.omega_zero
induction φ generalizing e with
| bounded φ => rfl
| conj φ ψ ihφ ihψ => exact (SigmaFormula.realize_conj _ _ _ _).trans (and_congr (ihφ e he) (ihψ e he))
| disj φ ψ ihφ ihψ => exact (SigmaFormula.realize_disj M h0 _ _ _).trans (or_congr (ihφ e he) (ihψ e he))
| existsSet φ ih =>
simp only [normalize,SigmaFormula.Realize,Realize]
apply exists_congr; intro x
apply and_congr_right; intro hx
exact ih (cons x e) (by intro i; cases i <;> simp [cons,he,hx])
| existsMem a φ ih =>
rw [normalize,SigmaFormula.realize_existsMem]
change (∃ x ∈ M, x ∈ e a ∧ _) ↔ ∃ x ∈ M, x ∈ e a ∧ _
apply exists_congr; intro x
apply and_congr_right; intro hx
exact and_congr_right (fun _ => ih (cons x e) (by intro i; cases i <;> simp [cons,he,hx]))
| allMem a φ ih =>
rw [normalize,SigmaFormula.realize_allMem M hM hP hU hS hR hI hC _ _ _ he]
change (∀ x ∈ e a, _) ↔ ∀ x ∈ M, x ∈ e a → _
constructor
· intro h x hx hxa
exact (ih (cons x e) (by intro i; cases i <;> simp [cons,he,hx])).mp (h x hxa)
· intro h x hxa
have hx := hM _ (he a) x hxa
exact (ih (cons x e) (by intro i; cases i <;> simp [cons,he,hx])).mpr (h x hx hxa)
end LevySigma
def LevySigmaSeparation (M : ZFSet.{u}) : Prop :=
∀ (φ : LevySigma) (e : ℕ → ZFSet.{u}), (∀ i, e i ∈ M) → ∀ a ∈ M,
∃ b ∈ M, ∀ x ∈ M, x ∈ b ↔ x ∈ a ∧ φ.Realize M (cons x e)
def LevySigmaReplacement (M : ZFSet.{u}) : Prop :=
∀ (φ : LevySigma) (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))
end TuringRigidity.BoundedSetTheory
end OAI
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