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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 | 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
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