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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.Computability.ArithmeticModelDecoding
namespace OAI
namespace TuringRigidity.BoundedDecoding
open SetCoding RelationCoding ArithmeticModelDecoding
theorem AntichainBelow.mono {p : AntichainCode} {x y : Degree} (hp : AntichainBelow p x) (h : x ≤ y) :
AntichainBelow p y := ⟨hp.1.trans h,hp.2.1.trans h,hp.2.2.trans h⟩
theorem SetBelow.mono {p : SetCode} {x y : Degree} (hp : SetBelow p x) (h : x ≤ y) :
SetBelow p y := ⟨hp.1.trans h,hp.2.1.mono h,hp.2.2.mono h⟩
theorem RelationBelow.mono {n : ℕ} {p : RelationCode n} {x y : Degree}
(hp : RelationBelow p x) (h : x ≤ y) : RelationBelow p y :=
⟨fun i => (hp.1 i).mono h,hp.2.mono h⟩
def unary (p : SetCode) : RelationCode 1 := ⟨fun _ => p,p.tags⟩
theorem unary_below {p : SetCode} {y : Degree} (hp : SetBelow p y) : RelationBelow (unary p) y :=
⟨fun _ => hp,hp.2.1⟩
theorem unary_holds (p : SetCode) (x : Degree) : (unary p).Holds (one x) ↔ p.Holds x := by
constructor
· rintro ⟨c,hc,_⟩
exact ⟨c 0,hc 0⟩
· rintro ⟨c,hc⟩
refine ⟨fun _ => c,fun i => ?_,?_⟩
· fin_cases i
exact hc
· simpa only [Finset.sup_const,Finset.univ_nonempty,ite_eq_left,unary] using hc.2.1
end TuringRigidity.BoundedDecoding
end OAI
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