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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 | 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
end OAI
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