| import OAI.Computability.DegreeRigidity.SetModels.RelationPresentation |
| import OAI.Computability.DegreeRigidity.Computability.OracleJump |
|
|
| namespace OAI |
|
|
| namespace TuringRigidity.DecodingTheorem |
| open IndexPresentation IndexTuples ArithmeticHierarchy RelationPresentation |
| open SetCoding RelationCoding BoundedDecoding OracleJump |
|
|
| def Presented {n} (Y : Oracle) (p : RelationCode n) (t : ℕ) : Prop := |
| (∀ i : Fin n, Dom Y (entry i.val t)) ∧ p.Holds (fun i => value Y (entry i.val t)) |
|
|
| theorem presentation_exact {n} (Y : Oracle) (p : RelationCode n) |
| (hp : RelationBelow p (degree Y)) (v : Fin n → Degree) : |
| p.Holds v ↔ ∃ t, Presented Y p t ∧ ∀ i, value Y (entry i.val t) = v i := by |
| constructor |
| · intro hv |
| have hb : ∀ i, v i ≤ degree Y := by |
| obtain ⟨c,hc,_⟩ := hv |
| exact fun i => (graph_bounded (hp.1 i) (hc i)).1 |
| choose e he hev using (fun i => value_surjective Y (v i) (hb i)) |
| obtain ⟨t,ht⟩ := tuple_surjective e |
| have hv' : (fun i : Fin n => value Y (entry i.val t)) = v := by |
| funext i |
| rw [ht i, hev i] |
| exact ⟨t, ⟨fun i => (ht i) ▸ he i, hv' ▸ hv⟩, fun i => congrFun hv' i⟩ |
| · rintro ⟨t,⟨_,ht⟩,hv⟩ |
| exact (funext hv) ▸ ht |
|
|
| theorem sigma5_normal_form {Y P} (h : Sigma Y 5 P) : |
| ∃ M : ℕ → Prop, RecursivePred Y M ∧ ∀ x, |
| P x ↔ ∃ a, ∀ b, ∃ c, ∀ d, ∃ e, |
| M (Nat.pair (Nat.pair (Nat.pair (Nat.pair (Nat.pair x a) b) c) d) e) := by |
| obtain ⟨A,hA,eA⟩ := h |
| obtain ⟨B,hB,eB⟩ := hA |
| obtain ⟨C,hC,eC⟩ := hB |
| obtain ⟨D,hD,eD⟩ := hC |
| obtain ⟨M,hM,eM⟩ := hD |
| refine ⟨M,hM,fun x => ?_⟩ |
| simp only [eA, eB, eC, eD, eM, Quant, Bool.not_true, Bool.not_false, Bool.false_eq_true, ↓reduceIte] |
|
|
| theorem decoding_theorem {n} (Y : Oracle) (R : Set (Fin n → Degree)) (p : RelationCode n) |
| (hp : RelationBelow p (degree Y)) (hcode : ∀ v, p.Holds v ↔ v ∈ R) : |
| ∃ P : ℕ → Prop, Sigma Y 5 P ∧ RecursivePred (iterate Y 5) P ∧ |
| ∀ v, v ∈ R ↔ ∃ t, P t ∧ ∀ i, value Y (entry i.val t) = v i := by |
| have h := sigma5_decoding Y p hp |
| refine ⟨Presented Y p, h, form_recursive h, fun v => ?_⟩ |
| exact (hcode v).symm.trans (presentation_exact Y p hp v) |
|
|
| end TuringRigidity.DecodingTheorem |
|
|
| end OAI |
|
|
