| 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 |
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|
| def NoDisagreement (A : ℕ → Oracle) (Y : Oracle) (e₀ e₁ : OracleCode) (p : Condition) : Prop := |
| ∀ q, Extends A p q → ¬ Disagreement Y e₀ e₁ q |
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| 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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