| import OAI.Computability.DegreeRigidity.Effective.UniformOracle | |
| namespace OAI | |
| namespace TuringRigidity.OracleCode | |
| theorem eval_mono {g h : ℕ →. ℕ} | |
| (hgh : ∀ n a, a ∈ g n → a ∈ h n) (c : OracleCode) (n a : ℕ) | |
| (ha : a ∈ eval g c n) : a ∈ eval h c n := by | |
| have happrox : ∀ k, Approximates (h k) (fun m => if m = 0 then g k else h k) := by | |
| intro k | |
| constructor | |
| · intro m b hb | |
| dsimp only at hb | |
| split at hb | |
| · exact hgh k b hb | |
| · exact hb | |
| · intro b hb | |
| exact ⟨1, fun m hm => by simpa [show m ≠ 0 by omega] using hb⟩ | |
| exact (eval_approximates happrox c n).1 0 a (by simpa using ha) | |
| theorem eval_finite_use (g : ℕ → ℕ) (c : OracleCode) (n a : ℕ) | |
| (ha : a ∈ eval (fun k => Part.some (g k)) c n) : | |
| ∃ m, a ∈ eval (BranchMachine.lookup (UniformOracle.oraclePrefix g m)) c n := by | |
| obtain ⟨m,hm⟩ := (eval_approximates (UniformOracle.prefix_approximates g) c n).2 a ha | |
| exact ⟨m,hm m le_rfl⟩ | |
| theorem eval_prefix_stable (g : ℕ → ℕ) (h : ℕ →. ℕ) (m : ℕ) | |
| (he : ∀ k, k < m → g k ∈ h k) (c : OracleCode) (n a : ℕ) | |
| (ha : a ∈ eval (BranchMachine.lookup (UniformOracle.oraclePrefix g m)) c n) : | |
| a ∈ eval h c n := by | |
| apply eval_mono (c := c) (n := n) (a := a) ?_ ha | |
| intro k b hb | |
| change b ∈ ((UniformOracle.oraclePrefix g m)[k]? : Part ℕ) at hb | |
| have hk : k < m := by | |
| by_contra hn | |
| have hnone : (UniformOracle.oraclePrefix g m)[k]? = none := | |
| List.getElem?_eq_none (by simpa [UniformOracle.oraclePrefix] using Nat.le_of_not_gt hn) | |
| simp [hnone] at hb | |
| have hv : (UniformOracle.oraclePrefix g m)[k]? = some (g k) := by | |
| simp [UniformOracle.oraclePrefix, List.getElem?_range hk] | |
| have heq : b = g k := by simpa [hv] using hb | |
| subst b | |
| exact he k hk | |
| theorem eval_total_stable (g : ℕ → ℕ) (c : OracleCode) (n a : ℕ) | |
| (ha : a ∈ eval (fun k => Part.some (g k)) c n) : | |
| ∃ m, ∀ h : ℕ → ℕ, (∀ k, k < m → h k = g k) → | |
| a ∈ eval (fun k => Part.some (h k)) c n := by | |
| obtain ⟨m,hm⟩ := eval_finite_use g c n a ha | |
| exact ⟨m,fun h hh => eval_prefix_stable g _ m | |
| (fun k hk => Part.mem_some_iff.mpr (hh k hk).symm) c n a hm⟩ | |
| end TuringRigidity.OracleCode | |
| end OAI | |