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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 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | 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
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