| 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⟩ |
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| 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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|
