File size: 4,341 Bytes
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 | import OAI.Computability.DegreeRigidity.Effective.CodingForcing
import OAI.Computability.DegreeRigidity.Representation.CommonIdealSearch
namespace OAI
namespace TuringRigidity.CodingAgreement
open CodingForcing
def Disagreement (Y : Oracle) (e₀ e₁ : OracleCode) (p : Condition) : Prop :=
∃ n a b, a ≠ b ∧ a ∈ CommonIdeal.run Y e₀ p.left n ∧ b ∈ CommonIdeal.run Y e₁ p.right n
def NoDisagreement (A : ℕ → Oracle) (Y : Oracle) (e₀ e₁ : OracleCode) (p : Condition) : Prop :=
∀ q, Extends A p q → ¬ Disagreement Y e₀ e₁ q
theorem disagreement_mono {A : ℕ → Oracle} {Y : Oracle} {e₀ e₁ : OracleCode}
{p q : Condition} (h : Extends A p q) (hd : Disagreement Y e₀ e₁ p) :
Disagreement Y e₀ e₁ q := by
obtain ⟨n, a, b, hab, ha, hb⟩ := hd
exact ⟨n, a, b, hab, CommonIdeal.run_mono h.1 n a ha, CommonIdeal.run_mono h.2.1 n b hb⟩
theorem disagreement_decision_dense (A : ℕ → Oracle) (Y : Oracle) (e₀ e₁ : OracleCode)
(p : Condition) :
∃ q, Extends A p q ∧ (Disagreement Y e₀ e₁ q ∨ NoDisagreement A Y e₀ e₁ q) := by
by_cases h : ∃ q, Extends A p q ∧ Disagreement Y e₀ e₁ q
· obtain ⟨q, hpq, hd⟩ := h
exact ⟨q, hpq, Or.inl hd⟩
· exact ⟨p, extends_refl A p, Or.inr (fun q hpq hd => h ⟨q, hpq, hd⟩)⟩
theorem noDisagreement_of_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 : Disagreement Y e₀ e₁ p ∨ NoDisagreement A Y e₀ e₁ p) :
NoDisagreement A Y e₀ e₁ p := by
rcases h with hd | hn
· obtain ⟨n, a, b, hab, ha, hb⟩ := hd
have hta := CommonIdeal.run_total h₀ n a ha
have htb := CommonIdeal.run_total h₁ n b hb
rw [he₀] at hta
rw [he₁] at htb
exact False.elim (hab (Part.mem_unique hta htb))
· exact hn
def UniqueValues (A : ℕ → Oracle) (Y : Oracle) (e : OracleCode) (p : Condition) : Prop :=
∀ q r, Extends A p q → Extends A p r → ∀ n a b,
a ∈ CommonIdeal.run Y e q.left n → b ∈ CommonIdeal.run Y e r.left n → a = b
theorem unique_search_sound {A : ℕ → Oracle} {Y G Z : Oracle} {e : OracleCode}
{p : Condition} (hu : UniqueValues A Y e p) (hG : CommonIdeal.Extends G p.left)
(he : OracleCode.eval (oracleFunction (join Y G)) e = oracleFunction Z)
(w : List Bool) (n a : ℕ) (ha : a ∈ CommonIdeal.run Y e (p.left ++ w) n) :
a = CommonIdeal.bit (Z n) := by
obtain ⟨q, hpq, hq⟩ := arbitrary_left_extension A p (p.left ++ w) (List.prefix_append _ _)
obtain ⟨s, hs, hb⟩ := CommonIdeal.run_finite_extension hG n (CommonIdeal.bit (Z n)) (by
rw [he]
exact Part.mem_some _)
obtain ⟨r, hpr, hr⟩ := arbitrary_left_extension A p s hs
exact hu q r hpq hpr n a (CommonIdeal.bit (Z n)) (hq ▸ ha) (hr ▸ hb)
theorem case1_reduces {A : ℕ → Oracle} {Y G Z : Oracle} {e : OracleCode} {p : Condition}
(hu : UniqueValues A Y e p) (hG : CommonIdeal.Extends G p.left)
(he : OracleCode.eval (oracleFunction (join Y G)) e = oracleFunction Z) : Reduces Z Y := by
obtain ⟨d, hd⟩ := CommonIdeal.finiteRun_code e p.left
apply RecursiveIn.iff_nat.mpr
apply UniformOracle.total_search (CommonIdeal.searchTrial_recursive Y d) (fun n => CommonIdeal.bit (Z n))
· intro n z a ha
have hf := Nat.Partrec.Code.evaln_sound ha
rw [hd] at hf
have hr := (CommonIdeal.finiteRun_approximates Y e p.left
(Nat.unpair (Nat.unpair z).2).1 n).1 (Nat.unpair z).1 a hf
exact unique_search_sound hu hG he _ n a hr
· intro n
obtain ⟨w, a, ha⟩ := CommonIdeal.agreeing_search_complete hG he n
obtain ⟨wcode, hw⟩ := BorelGeneric.word_surjective w
rw [← hw] at ha
obtain ⟨m, hm⟩ := (CommonIdeal.finiteRun_approximates Y e p.left wcode n).2 a ha
have hf := hm m le_rfl
dsimp only at hf
rw [← hd] at hf
obtain ⟨t, ht⟩ := Nat.Partrec.Code.evaln_complete.mp hf
exact ⟨Nat.pair m (Nat.pair wcode t), a, by simpa [CommonIdeal.searchTrial] using ht⟩
end TuringRigidity.CodingAgreement
end OAI
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