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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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | import OAI.Computability.DegreeRigidity.Effective.CodingExtraction
namespace OAI
namespace TuringRigidity.CodingRequirements
open CodingForcing CodingAgreement CodingLocations CodingExtraction
def Good (A : ℕ → Oracle) (Y : Oracle) (e₀ e₁ : OracleCode) (p : Condition) : Prop :=
Disagreement Y e₀ e₁ p ∨
(∃ n, ∀ q, Extends A p q → ∀ v, v ∉ CommonIdeal.run Y e₀ q.left n) ∨
(∃ n, ∀ q, Extends A p q → ∀ v, v ∉ CommonIdeal.run Y e₁ q.right n) ∨
UniqueValues A Y e₀ p ∨ ∃ k, Reduces (A k) Y
theorem good_dense (A : ℕ → Oracle)
(hIntro : ∀ k Z, (∀ a, Z a = true → A k a = true) →
{a | Z a = true}.Infinite → Reduces (A k) Z)
(Y : Oracle) (e₀ e₁ : OracleCode) (p : Condition) :
∃ q, Extends A p q ∧ Good A Y e₀ e₁ q := by
obtain ⟨q, hpq, hd | hn⟩ := disagreement_decision_dense A Y e₀ e₁ p
· exact ⟨q, hpq, Or.inl hd⟩
· by_cases hl : ∃ r, Extends A q r ∧ ∃ n, ∀ s, Extends A r s → ∀ v, v ∉ CommonIdeal.run Y e₀ s.left n
· obtain ⟨r, hqr, hdiv⟩ := hl
exact ⟨r, extends_trans hpq hqr, Or.inr (Or.inl hdiv)⟩
· by_cases hr : ∃ r, Extends A q r ∧ ∃ n, ∀ s, Extends A r s → ∀ v, v ∉ CommonIdeal.run Y e₁ s.right n
· obtain ⟨r, hqr, hdiv⟩ := hr
exact ⟨r, extends_trans hpq hqr, Or.inr (Or.inr (Or.inl hdiv))⟩
· by_cases hu : ∃ r, Extends A q r ∧ UniqueValues A Y e₀ r
· obtain ⟨r, hqr, hu⟩ := hu
exact ⟨r, extends_trans hpq hqr, Or.inr (Or.inr (Or.inr (Or.inl hu)))⟩
· have hL : DenseLeft A Y e₀ q := by
intro r hqr n
by_contra h
push Not at h
exact hl ⟨r, hqr, n, fun s hrs v => h s v hrs⟩
have hR : DenseRight A Y e₁ q := by
intro r hqr n
by_contra h
push Not at h
exact hr ⟨r, hqr, n, fun s hrs v => h s v hrs⟩
have hU : ∀ r, Extends A q r → ¬ UniqueValues A Y e₀ r :=
fun r hqr huni => hu ⟨r, hqr, huni⟩
obtain ⟨k, _, hk⟩ := case2_computes_member hIntro hn hL hR hU
exact ⟨q, hpq, Or.inr (Or.inr (Or.inr (Or.inr ⟨k, hk⟩)))⟩
theorem arbitrary_right_extension (A : ℕ → Oracle) (p : Condition)
(s : List Bool) (hs : p.right <+: s) : ∃ q, Extends A p q ∧ q.right = s := by
obtain ⟨u, rfl⟩ := hs
exact ⟨append p u, append_extends A p u, rfl⟩
theorem good_common_output {A : ℕ → Oracle} {Y G₀ G₁ Z : Oracle}
{e₀ e₁ : OracleCode} {p : Condition}
(h₀ : CommonIdeal.Extends G₀ p.left) (h₁ : CommonIdeal.Extends G₁ p.right)
(he₀ : OracleCode.eval (oracleFunction (join Y G₀)) e₀ = oracleFunction Z)
(he₁ : OracleCode.eval (oracleFunction (join Y G₁)) e₁ = oracleFunction Z)
(h : Good A Y e₀ e₁ p) : Reduces Z Y ∨ ∃ k, Reduces (A k) Y := by
rcases h with hd | hl | hr | hu | ha
· obtain ⟨n, a, b, hab, hva, hvb⟩ := hd
have hta := CommonIdeal.run_total h₀ n a hva
have htb := CommonIdeal.run_total h₁ n b hvb
rw [he₀] at hta
rw [he₁] at htb
exact False.elim (hab (Part.mem_unique hta htb))
· obtain ⟨n, hn⟩ := hl
obtain ⟨s, hs, hv⟩ := CommonIdeal.run_finite_extension h₀ n (CommonIdeal.bit (Z n)) (by
rw [he₀]; exact Part.mem_some _)
obtain ⟨q, hpq, hq⟩ := arbitrary_left_extension A p s hs
exact False.elim (hn q hpq _ (hq ▸ hv))
· obtain ⟨n, hn⟩ := hr
obtain ⟨s, hs, hv⟩ := CommonIdeal.run_finite_extension h₁ n (CommonIdeal.bit (Z n)) (by
rw [he₁]; exact Part.mem_some _)
obtain ⟨q, hpq, hq⟩ := arbitrary_right_extension A p s hs
exact False.elim (hn q hpq _ (hq ▸ hv))
· exact Or.inl (case1_reduces hu h₀ he₀)
· exact Or.inr ha
def GoodForCode (A : ℕ → Oracle) (B : Oracle) (c e₀ e₁ : OracleCode) (p : Condition) : Prop :=
∀ Y : Oracle, OracleCode.eval (oracleFunction B) c = oracleFunction Y → Good A Y e₀ e₁ p
theorem goodForCode_dense (A : ℕ → Oracle)
(hIntro : ∀ k Z, (∀ a, Z a = true → A k a = true) →
{a | Z a = true}.Infinite → Reduces (A k) Z)
(B : Oracle) (c e₀ e₁ : OracleCode) (p : Condition) :
∃ q, Extends A p q ∧ GoodForCode A B c e₀ e₁ q := by
by_cases h : ∃ Y : Oracle, OracleCode.eval (oracleFunction B) c = oracleFunction Y
· obtain ⟨Y, hY⟩ := h
obtain ⟨q, hpq, hg⟩ := good_dense A hIntro Y e₀ e₁ p
refine ⟨q, hpq, ?_⟩
intro Z hZ
have he : Y = Z := oracleFunction_injective (hY.symm.trans hZ)
exact he ▸ hg
· exact ⟨p, extends_refl A p, fun Y hY => False.elim (h ⟨Y, hY⟩)⟩
end TuringRigidity.CodingRequirements
end OAI
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