| import OAI.Computability.DegreeRigidity.Model | |
| import Mathlib.Computability.TuringDegree | |
| import Mathlib.Order.Hom.Basic | |
| namespace OAI | |
| namespace TuringRigidity | |
| theorem reduces_refl (A : Oracle) : Reduces A A := TuringReducible.refl _ | |
| theorem reduces_trans {A B C : Oracle} (h : Reduces A B) (k : Reduces B C) : | |
| Reduces A C := TuringReducible.trans h k | |
| def degree (A : Oracle) : Degree := toAntisymmetrization Reduces A | |
| @[simp] theorem degree_le_iff (A B : Oracle) : degree A ≤ degree B ↔ Reduces A B := Iff.rfl | |
| theorem degree_eq_iff (A B : Oracle) : | |
| degree A = degree B ↔ Reduces A B ∧ Reduces B A := by | |
| exact Quotient.eq | |
| theorem degree_surjective : Function.Surjective degree := by | |
| intro a | |
| exact ⟨ofAntisymmetrization Reduces a, toAntisymmetrization_ofAntisymmetrization _ a⟩ | |
| theorem order_rigidity_of_inverse_le {α : Type*} [PartialOrder α] | |
| (h : ∀ π : α ≃o α, ∀ a, π.symm a ≤ a) : | |
| ∀ π : α ≃o α, ∀ a, π a = a := by | |
| intro π a | |
| apply le_antisymm | |
| · simpa using h π.symm a | |
| · simpa using π.monotone (h π a) | |
| end TuringRigidity | |
| end OAI | |