| import OAI.Computability.DegreeRigidity.SetModels.ModelDegreeCodes |
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| namespace OAI |
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| namespace TuringRigidity.BoundedSetTheory |
| universe u |
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| 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) |
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| 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)))) |
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| 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 |
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| end TuringRigidity.BoundedSetTheory |
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| end OAI |
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