openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /PersistentExtension.lean
| 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 | |