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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 | 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
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