openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Representation /GenericOneRealNames.lean
| import OAI.Computability.DegreeRigidity.Constructibility.PersistentOneRealDefinition | |
| import OAI.Computability.DegreeRigidity.Syntax.SetSatisfactionBounded | |
| import OAI.Computability.DegreeRigidity.Constructibility.PersistentExtensionName | |
| namespace OAI | |
| namespace TuringRigidity.FullSetForcing | |
| open TransitiveNameModel BoundedSetTheory ElementaryModel SetDegreeDecoding | |
| open PersistentRestrictions RecursiveNames CountableForcing AtomicForcing RelativeConstructible | |
| universe u | |
| theorem persistent_extension_one_real_names | |
| (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 ∧ | |
| ∃ (δ : Ordinal.{u}) (P : Oracle) (p : SentenceForm), | |
| δ.toZFSet ∈ E ∧ P ∈ modelReals E ∧ p.bound ≤ 2 ∧ | |
| realCode P ∈ level (groundReals E) δ ∧ | |
| automorphismSet σ = definedSubset (level (groundReals E) δ) p | |
| (fun _ : Fin p.bound => realCode P) ∧ | |
| ∃ r d a l f : Name (Conditions c), | |
| r.encode (label c) ∈ M ∧ d.encode (label c) ∈ M ∧ | |
| a.encode (label c) ∈ M ∧ l.encode (label c) ∈ M ∧ f.encode (label c) ∈ M ∧ | |
| r.val G.carrier = groundReals E ∧ d.val G.carrier = δ.toZFSet ∧ | |
| a.val G.carrier = realCode P ∧ l.val G.carrier = level (groundReals E) δ ∧ | |
| f.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,δ,P,p,hδ,hP,hp,hPL,hdef⟩ := | |
| persistent_extension_one_real_definition E hE hTE I ρ hρ hz hIJ | |
| have hrE := groundReals_mem E hE hTE | |
| have hlE := level_mem E _ hE hTE hrE δ hδ | |
| have hfE : automorphismSet σ ∈ E := by | |
| rw [hdef] | |
| exact definedSubset_mem E _ hE hTE.separation.finitePrefix.bounded hlE p | |
| (fun _ => realCode P) (fun _ => hPL) (by omega) | |
| obtain ⟨r,hr,hvr⟩ := (mem_extensionSet M c G.carrier _).mp hrE | |
| obtain ⟨d,hd,hvd⟩ := (mem_extensionSet M c G.carrier _).mp hδ | |
| obtain ⟨a,ha,hva⟩ := (mem_extensionSet M c G.carrier (realCode P)).mp hP | |
| obtain ⟨l,hl,hvl⟩ := (mem_extensionSet M c G.carrier _).mp hlE | |
| obtain ⟨f,hf,hvf⟩ := (mem_extensionSet M c G.carrier _).mp hfE | |
| exact ⟨hIJ,σ,he,hσ,δ,P,p,hδ,hP,hp,hPL,hdef,r,d,a,l,f, | |
| hr,hd,ha,hl,hf,hvr,hvd,hva,hvl,hvf⟩ | |
| end TuringRigidity.FullSetForcing | |
| end OAI | |