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