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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 | import OAI.Computability.DegreeRigidity.SetModels.ModelDegreeCodes
namespace OAI
namespace TuringRigidity.BoundedSetTheory
universe u
def DegreeEqualityFormula (φ : Formula) : Prop := ∀ o Q z x y : ℕ, ∀ e : ℕ → ZFSet.{u},
e o = ZFSet.omega → (∀ f : ℕ → ℕ, ∀ k, finiteNaturalGraph f k ∈ e Q) →
e z = natSet 0 → ∀ A B : Oracle, e x = realCode A → e y = realCode B →
((Formula.twoOracles φ o Q z x y).Eval e ↔ degree A = degree B)
namespace Formula
def degreeSet (φ : Formula) (o Q z R A D : ℕ) : Formula :=
.conj (.subset D R) (allMem R (iff (.member 0 (D+1)) (twoOracles φ (o+1) (Q+1) (z+1) 0 (A+1))))
theorem degreeSet_spec {φ : Formula} (hφ : DegreeEqualityFormula.{u} φ)
(o Q z R x D : ℕ) (e : ℕ → ZFSet.{u}) (ho : e o = ZFSet.omega)
(hQ : ∀ f : ℕ → ℕ, ∀ k, finiteNaturalGraph f k ∈ e Q) (hz : e z = natSet 0)
(hR : ∀ w ∈ e R, ∃ B : Oracle, realCode B = w)
(A : Oracle) (hA : e x = realCode A) :
(degreeSet φ o Q z R x D).Eval e ↔ e D = degreeCode (e R) A := by
simp only [degreeSet,Formula.Eval,eval_subset,eval_allMem,eval_iff,cons_zero,cons_succ]
constructor
· rintro ⟨hsub,h⟩
apply ZFSet.ext; intro w
by_cases hw : w ∈ e R
· obtain ⟨B,rfl⟩ := hR w hw
rw [realCode_mem_degreeCode]
have hh := (h _ hw).trans (hφ (o+1) (Q+1) (z+1) 0 (x+1) (cons (realCode B) e)
ho hQ hz B A rfl hA)
exact hh.trans (and_iff_right hw).symm
· exact ⟨fun h => False.elim (hw (hsub h)),fun h => False.elim (hw (degreeCode_subset _ _ h))⟩
· intro hd
refine ⟨hd.symm ▸ degreeCode_subset (e R) A,?_⟩
intro w hw
obtain ⟨B,rfl⟩ := hR w hw
rw [hd,realCode_mem_degreeCode,and_iff_right hw]
exact (hφ (o+1) (Q+1) (z+1) 0 (x+1) (cons (realCode B) e) ho hQ hz B A rfl hA).symm
end Formula
end TuringRigidity.BoundedSetTheory
end OAI
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