openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /PersistentExtensionName.lean
| import OAI.Computability.DegreeRigidity.Constructibility.RelativePersistentExtension | |
| import OAI.Computability.DegreeRigidity.Representation.CommonGenericModels | |
| namespace OAI | |
| namespace TuringRigidity.FullSetForcing | |
| open TransitiveNameModel BoundedSetTheory ElementaryModel SetDegreeDecoding | |
| open PersistentRestrictions RecursiveNames CountableForcing AtomicForcing | |
| universe u | |
| theorem persistent_extension_name (M : ZFSet.{u}) [Countable (Conditions M)] | |
| (hM : Transitive M) (hT : SourceT M) | |
| {c o : ZFSet.{u}} [Preorder (Conditions c)] [Top (Conditions c)] | |
| (hc : c ∈ M) (hoM : o ∈ M) | |
| (ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s) | |
| (G : GenericFilter (Conditions c)) (hG : GroundGeneric M G) (htop : ⊤ ∈ G.carrier) | |
| (I : CountableIdeal) (ρ : I ≃o I) (hρ : Persistent I ρ) | |
| (hz : degree (OracleJump.jump FixedArithmetic.zero) ∈ I.carrier) | |
| (hIM : I.carrier ⊆ (modelIdeal M hM hT).carrier) : | |
| let E := genericExtensionSet M c G.carrier | |
| let hE := genericExtensionSet_transitive M c hM G.carrier | |
| let hTE := extension_sourceT M hM hT hc hoM ho G hG htop | |
| let J := modelIdeal E hE hTE | |
| ∃ hIJ : I.carrier ⊆ J.carrier, ∃ σ : J ≃o J, | |
| Extends hIJ ρ σ ∧ Persistent J σ ∧ | |
| ∃ t : Name (Conditions c), t.encode (label c) ∈ M ∧ t.val G.carrier = automorphismSet σ := by | |
| intro E hE hTE J | |
| have hME := ground_inclusion_set M c hM hT.pairing hT.union hT.powerSet | |
| hT.separation.finitePrefix.bounded hT.replacement.finitePrefix hT.infinity hc G.carrier htop | |
| have hIJ : I.carrier ⊆ J.carrier := by | |
| intro d hd | |
| obtain ⟨A,hA,rfl⟩ := hIM hd | |
| refine ⟨A,?_,rfl⟩ | |
| change realCode A ∈ E | |
| exact hME hA | |
| obtain ⟨σ,he,hσ,hσE⟩ := RelativeConstructible.persistent_extension_in_model E hE hTE I ρ hρ hz hIJ | |
| obtain ⟨t,ht,hval⟩ := (mem_extensionSet M c G.carrier _).mp hσE | |
| exact ⟨hIJ,σ,he,hσ,t,ht,hval⟩ | |
| theorem model_automorphism_output_name (M c : ZFSet.{u}) [Countable (Conditions M)] | |
| (G : Set (Conditions c)) | |
| (hE : Transitive (genericExtensionSet M c G)) (hTE : SourceT (genericExtensionSet M c G)) | |
| (σ : modelIdeal (genericExtensionSet M c G) hE hTE ≃o | |
| modelIdeal (genericExtensionSet M c G) hE hTE) | |
| (A : Oracle) (hA : A ∈ modelReals (genericExtensionSet M c G)) : | |
| ∃ B : Oracle, ∃ t : Name (Conditions c), | |
| t.encode (label c) ∈ M ∧ t.val G = realCode B ∧ | |
| degree B = (σ ⟨degree A,⟨A,hA,rfl⟩⟩).val := by | |
| obtain ⟨B,hB,hdegree⟩ := (σ ⟨degree A,⟨A,hA,rfl⟩⟩).property | |
| obtain ⟨t,ht,hval⟩ := (mem_extensionSet M c G (realCode B)).mp hB | |
| exact ⟨B,t,ht,hval,hdegree⟩ | |
| end TuringRigidity.FullSetForcing | |
| end OAI | |