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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 | 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
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