| import OAI.Computability.DegreeRigidity.Effective.CodingSplitting |
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| namespace OAI |
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| namespace TuringRigidity.CodingLocations |
| open CodingForcing CodingAgreement CodingOnePoint CodingSplitting |
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| def DenseLeft (A : ℕ → Oracle) (Y : Oracle) (e : OracleCode) (p : Condition) : Prop := |
| ∀ q, Extends A p q → ∀ n, ∃ r v, Extends A q r ∧ v ∈ CommonIdeal.run Y e r.left n |
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| def DenseRight (A : ℕ → Oracle) (Y : Oracle) (e : OracleCode) (p : Condition) : Prop := |
| ∀ q, Extends A p q → ∀ n, ∃ r v, Extends A q r ∧ v ∈ CommonIdeal.run Y e r.right n |
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| def SplittingLocation (Y : Oracle) (e : OracleCode) (base : List Bool) (m : ℕ) : Prop := |
| ∃ s b n v w, base <+: s ∧ base.length ≤ m ∧ m < s.length ∧ v ≠ w ∧ |
| v ∈ CommonIdeal.run Y e s n ∧ w ∈ CommonIdeal.run Y e (s.set m b) n |
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| theorem splittingLocation_sound {A : ℕ → Oracle} {Y : Oracle} {e₀ e₁ : OracleCode} |
| {p : Condition} (hn : NoDisagreement A Y e₀ e₁ p) (hR : DenseRight A Y e₁ p) |
| {m : ℕ} (hm : SplittingLocation Y e₀ p.left m) : CodingLocation A p m := by |
| obtain ⟨s, b, n, v, w, hs, hmp, hms, hvw, hv, hw⟩ := hm |
| obtain ⟨r, hpr, hr⟩ := arbitrary_left_extension A p s hs |
| have hms' : m < r.left.length := by simpa only [hr] using hms |
| rcases split_coding_or_divergence hn hpr m hmp hms' b n v w hvw |
| (hr ▸ hv) (hr ▸ hw) with hc | hd |
| · exact hc |
| · obtain ⟨q, c, hrq, hc⟩ := hR r hpr n |
| exact False.elim (hd q hrq c hc) |
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| theorem splittingLocation_unbounded {A : ℕ → Oracle} {Y : Oracle} {e : OracleCode} |
| {p : Condition} (hL : DenseLeft A Y e p) |
| (hU : ∀ q, Extends A p q → ¬ UniqueValues A Y e q) (N : ℕ) : |
| ∃ m, N ≤ m ∧ SplittingLocation Y e p.left m := by |
| obtain ⟨q, hpq, hNq⟩ := length_dense A p N |
| have hu := hU q hpq |
| simp only [UniqueValues] at hu |
| push Not at hu |
| obtain ⟨r, s, hqr, hqs, n, a, b, ha, hb, hab⟩ := hu |
| rcases split_or_divergent_word hqr.1 hqs.1 n a b ha hb hab with hsplit | hdiv |
| · obtain ⟨w, m, c, v, z, hqw, hqm, hmw, hvz, hv, hz⟩ := hsplit |
| exact ⟨m, hNq.trans hqm, w, c, n, v, z, hpq.1.trans hqw, |
| hpq.1.length_le.trans hqm, hmw, hvz, hv, hz⟩ |
| · obtain ⟨w, hqw, hd⟩ := hdiv |
| obtain ⟨t, hqt, ht⟩ := arbitrary_left_extension A q w hqw |
| obtain ⟨u, v, htu, hv⟩ := hL t (extends_trans hpq hqt) n |
| have hwu : w <+: u.left := by simpa only [ht] using htu.1 |
| obtain ⟨extra, hleft⟩ := hwu |
| exact False.elim (hd extra v (hleft ▸ hv)) |
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| theorem splittingLocations_infinite {A : ℕ → Oracle} {Y : Oracle} {e : OracleCode} |
| {p : Condition} (hL : DenseLeft A Y e p) |
| (hU : ∀ q, Extends A p q → ¬ UniqueValues A Y e q) : |
| {m | SplittingLocation Y e p.left m}.Infinite := by |
| intro hf |
| have hb := hf.bddAbove |
| obtain ⟨b, hb⟩ := hb |
| obtain ⟨m, hm, hs⟩ := splittingLocation_unbounded hL hU (b+1) |
| have hmb := hb hs |
| omega |
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| end TuringRigidity.CodingLocations |
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| end OAI |
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