File size: 2,959 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 | import OAI.Computability.DegreeRigidity.Effective.CodingSplitting
namespace OAI
namespace TuringRigidity.CodingLocations
open CodingForcing CodingAgreement CodingOnePoint CodingSplitting
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
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
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
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)
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))
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
end TuringRigidity.CodingLocations
end OAI
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