openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /ParameterPersistentCollapse.lean
| import OAI.Computability.DegreeRigidity.Representation.ParameterCoveringModels | |
| import OAI.Computability.DegreeRigidity.Representation.GroundDegreeCollapse | |
| namespace OAI | |
| namespace TuringRigidity.BoundedSetTheory | |
| open TransitiveNameModel PersistentRestrictions PersistentPresentation PersistentCountability OracleJump | |
| open CountableForcing ElementaryModel | |
| universe u | |
| attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder | |
| theorem parameter_persistent_collapse (π : Degree ≃o Degree) (A : ℕ → Oracle) : | |
| ∃ M : ZFSet.{u}, Transitive M ∧ SourceT M ∧ Countable (Conditions M) ∧ | |
| (∀ n, A n ∈ modelReals M) ∧ | |
| ∃ c ∈ M, ∃ G : GenericFilter (Conditions c), AtomicForcing.GroundGeneric M G ∧ | |
| let N := genericExtensionSet M c G.carrier | |
| M ⊆ N ∧ Transitive N ∧ SourceT N ∧ | |
| ∃ I : CountableIdeal, ∃ H ∈ modelReals N, ∃ hH : Presented I H, | |
| (∀ a : Degree, a ∈ I.carrier ↔ ∃ X ∈ modelReals M, degree X = a) ∧ | |
| (∀ n, degree (A n) ∈ I.carrier) ∧ | |
| ∃ ρ : I ≃o I, RestrictsTo π I ρ ∧ Persistent I ρ ∧ | |
| graphOracle hH ρ ∈ modelReals N ∧ | |
| ∃ S ∈ N, degreeUniverse S ∈ N ∧ degreeOrder S ∈ N ∧ | |
| presentationGraph S H (graphOracle hH ρ) ∈ N ∧ | |
| (LevySigma.persistent 0 1 2 3 4).Realize N | |
| (modelDegreeEnv S (presentationSet S H) (presentationGraph S H (graphOracle hH ρ))) := by | |
| obtain ⟨X,hX⟩ := degree_surjective (π (degree (jump FixedArithmetic.zero))) | |
| obtain ⟨Y,hY⟩ := degree_surjective (π.symm (degree (jump FixedArithmetic.zero))) | |
| let B : ℕ → Oracle | 0 => X | 1 => Y | n+2 => A n | |
| obtain ⟨M,hM,hT,hcount,hB⟩ := parameter_covering_models.{u} B | |
| let : Countable (Conditions M) := hcount | |
| obtain ⟨c,hc,G,hG,hMN,hN,hTN,I,H,hHN,hH,hI,ρ,hρ,hp,hRN,S,hSN,hU,hO,hgraph⟩ := | |
| ground_degree_persistent_collapse M hM hT π X Y (hB 0) (hB 1) hX hY | |
| refine ⟨M,hM,hT,hcount,(fun n => hB (n+2)),c,hc,G,hG,hMN,hN,hTN, | |
| I,H,hHN,hH,hI,?_,ρ,hρ,hp,hRN,S,hSN,hU,hO,hgraph⟩ | |
| intro n | |
| exact (hI _).mpr ⟨A n,hB (n+2),rfl⟩ | |
| end TuringRigidity.BoundedSetTheory | |
| end OAI | |