| import OAI.Computability.DegreeRigidity.Effective.CodingColumns |
|
|
| namespace OAI |
|
|
| namespace TuringRigidity.CodingDiagonal |
| open CodingForcing CodingColumns Introreducible |
|
|
| def Diagonal (A : ℕ → Oracle) (hA : ∀ k, {a | A k a = true}.Infinite) |
| (k cutoff : ℕ) (e : OracleCode) (p : Condition) : Prop := |
| ∃ n, location (A k) (hA k) k cutoff n < p.left.length ∧ |
| OracleCode.eval (oracleFunction (A k)) e n ≠ |
| Part.some (if p.left.getD (location (A k) (hA k) k cutoff n) false then 1 else 0) |
|
|
| theorem diagonal_mono {A : ℕ → Oracle} {hA : ∀ k, {a | A k a = true}.Infinite} |
| {k cutoff : ℕ} {e : OracleCode} {p q : Condition} |
| (hpq : Extends A p q) (hp : Diagonal A hA k cutoff e p) : |
| Diagonal A hA k cutoff e q := by |
| obtain ⟨n, hn, he⟩ := hp |
| exact ⟨n, hn.trans_le hpq.1.length_le, by rwa [getD_of_prefix hpq.1 hn]⟩ |
|
|
| theorem exists_different_bit (v : Part ℕ) : |
| ∃ b : Bool, v ≠ Part.some (if b then 1 else 0) := by |
| by_cases h : v = Part.some 0 |
| · exact ⟨true, by simp [h]⟩ |
| · exact ⟨false, h⟩ |
|
|
| theorem prescribe (A : ℕ → Oracle) (p : Condition) (m : ℕ) |
| (hm : p.left.length ≤ m) (b : Bool) : |
| ∃ q, Extends A p q ∧ m < q.left.length ∧ q.left.getD m false = b := by |
| refine ⟨append p (List.replicate (m - p.left.length + 1) b), append_extends A p _, ?_, ?_⟩ |
| · simp only [append, List.length_append, List.length_replicate] |
| omega |
| · simp only [append, List.getD_eq_getElem?_getD, List.getElem?_append, |
| ite_eq_right (Nat.not_lt.mpr hm)] |
| simp |
|
|
| theorem diagonal_dense (A : ℕ → Oracle) (hA : ∀ k, {a | A k a = true}.Infinite) |
| (k cutoff : ℕ) (e : OracleCode) (p : Condition) : |
| ∃ q, Extends A p q ∧ Diagonal A hA k cutoff e q := by |
| let n := p.left.length |
| let m := location (A k) (hA k) k cutoff n |
| obtain ⟨b, hb⟩ := exists_different_bit (OracleCode.eval (oracleFunction (A k)) e n) |
| obtain ⟨q, hpq, hmq, hqb⟩ := prescribe A p m (le_location _ _ _ _ _) b |
| refine ⟨q, hpq, n, hmq, ?_⟩ |
| simpa only [← hqb] using hb |
|
|
| end TuringRigidity.CodingDiagonal |
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|
| end OAI |
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|
