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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 | import OAI.Computability.DegreeRigidity.Constructibility.PersistentRestrictionCode
namespace OAI
namespace TuringRigidity.PersistentExtension
open EncodedForcing OracleJump ArithmeticHierarchy IdealInterpretation IdealLocality
open PersistentRestrictions PersistentLocality PersistentPresentation PersistentCountability
noncomputable section
theorem nonpersistent_witness {J : CountableIdeal} {σ : J ≃o J}
(h : ¬ Persistent J σ) :
∃ x, ∀ (K : CountableIdeal) (hJK : J.carrier ⊆ K.carrier) (τ : K ≃o K),
x ∈ K.carrier → ¬ Extends hJK σ τ := by
simpa only [Persistent,not_forall,not_exists,not_and] using h
theorem source_4_1_10 (I J : CountableIdeal) (ρ : I ≃o I)
(hp : Persistent I ρ) (hz : degree (jump FixedArithmetic.zero) ∈ I.carrier)
(hIJ : I.carrier ⊆ J.carrier) (hj : JumpClosed (ideal J)) :
∃ σ : J ≃o J, Extends hIJ ρ σ ∧ Persistent J σ := by
classical
by_contra hn
have hnot (σ : J ≃o J) (he : Extends hIJ ρ σ) : ¬ Persistent J σ :=
fun hpσ => hn ⟨σ,he,hpσ⟩
obtain ⟨H,hpres⟩ := presentation_exists J
let S : Set (J ≃o J) := {σ | RecursivePred (iterate H 11) (Graph σ hpres)}
have hS : S.Countable := arithmetic_maps_countable hpres _
have : Countable S := hS.to_subtype
have hwitness (σ : S) : ∃ x, Extends hIJ ρ σ.val →
∀ (K : CountableIdeal) (hJK : J.carrier ⊆ K.carrier) (τ : K ≃o K),
x ∈ K.carrier → ¬ Extends hJK σ.val τ := by
by_cases he : Extends hIJ ρ σ.val
· obtain ⟨x,hx⟩ := nonpersistent_witness (hnot σ.val he)
exact ⟨x,fun _ => hx⟩
· exact ⟨⊥,fun h => (he h).elim⟩
choose w hw using hwitness
let L := JumpIdeal.generated (degree H)
have hJL : J.carrier ⊆ L.carrier := by
intro x hx
obtain ⟨n,rfl⟩ := (hpres x).mp hx
exact L.lower (CodingExtraction.column_projection_reduces H n) (JumpIdeal.includes _)
obtain ⟨b,hb⟩ := CountableBound.countable_bounded (Set.range w ∪ L.carrier)
((Set.countable_range w).union L.countable)
obtain ⟨K,hIK,τ,hbK,he⟩ := hp b
have hLK : L.carrier ⊆ K.carrier := fun x hx => K.lower (hb x (Or.inr hx)) hbK
have hJK : J.carrier ⊆ K.carrier := fun x hx => hLK (hJL hx)
obtain ⟨σ,hρ,hτ,hr⟩ := PersistentRestrictionCode.restriction ρ τ hIK he hz
hIJ hJK hj hpres hLK
let s : S := ⟨σ,hr⟩
have hwK : w s ∈ K.carrier := K.lower (hb (w s) (Or.inl ⟨s,rfl⟩)) hbK
exact hw s hρ K hJK τ hwK hτ
end
end TuringRigidity.PersistentExtension
end OAI
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