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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 37 38 39 | 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
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