| import Mathlib | |
| namespace OAI | |
| namespace TuringRigidity | |
| abbrev Oracle := ℕ → Bool | |
| def oracleFunction (A : Oracle) : ℕ →. ℕ := fun n => Part.some (if A n then 1 else 0) | |
| def Reduces (A B : Oracle) : Prop := TuringReducible (oracleFunction A) (oracleFunction B) | |
| instance : IsPreorder Oracle Reduces where | |
| refl A := TuringReducible.refl (oracleFunction A) | |
| trans _ _ _ h k := TuringReducible.trans h k | |
| def Degree := Antisymmetrization Oracle Reduces | |
| instance : PartialOrder Degree := | |
| @instPartialOrderAntisymmetrization Oracle | |
| { le := Reduces | |
| lt A B := Reduces A B ∧ ¬ Reduces B A | |
| lt_iff_le_not_ge _ _ := Iff.rfl | |
| le_refl A := TuringReducible.refl (oracleFunction A) | |
| le_trans _ _ _ h k := TuringReducible.trans h k } | |
| private instance oraclePreorder : Preorder Oracle where | |
| le := Reduces | |
| lt A B := Reduces A B ∧ ¬ Reduces B A | |
| lt_iff_le_not_ge _ _ := Iff.rfl | |
| le_refl A := TuringReducible.refl (oracleFunction A) | |
| le_trans _ _ _ h k := TuringReducible.trans h k | |
| def MainTheorem : Prop := ∀ π : Degree ≃o Degree, ∀ a : Degree, π a = a | |
| end TuringRigidity | |
| end OAI | |