| import OAI.Computability.DegreeRigidity.Effective.UniformOracle |
| import OAI.Computability.DegreeRigidity.Computability.Joins |
|
|
| namespace OAI |
|
|
| namespace TuringRigidity.CountableBound |
| open UniformOracle |
|
|
| def sum (A : ℕ → Oracle) : Oracle := fun n => A (Nat.unpair n).1 (Nat.unpair n).2 |
|
|
| theorem reduces_sum (A : ℕ → Oracle) (k : ℕ) : Reduces (A k) (sum A) := by |
| apply RecursiveIn.iff_nat.mpr |
| have hO : Nat.RecursiveIn {oracleFunction (sum A)} (oracleFunction (sum A)) := |
| .oracle _ (Set.mem_singleton _) |
| have h := total_comp hO (total_pair (total_primrec (Primrec.const k)) (total_primrec Primrec.id)) |
| exact h.of_eq (fun n => by simp [oracleFunction, sum]) |
|
|
| theorem countable_bounded (S : Set Degree) (hS : S.Countable) : |
| ∃ b : Degree, ∀ x ∈ S, x ≤ b := by |
| by_cases hne : S.Nonempty |
| · obtain ⟨f, hf⟩ := hS.exists_eq_range hne |
| choose A hA using (fun n => degree_surjective (f n)) |
| refine ⟨degree (sum A), ?_⟩ |
| intro x hx |
| rw [hf] at hx |
| obtain ⟨n, rfl⟩ := hx |
| rw [← hA n] |
| exact reduces_sum A n |
| · exact ⟨⊥, fun x hx => False.elim (hne ⟨x, hx⟩)⟩ |
|
|
| end TuringRigidity.CountableBound |
|
|
| end OAI |
|
|
