openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /PersistentConstruction.lean
| import OAI.Computability.DegreeRigidity.Constructibility.ConstructionDefinition | |
| import OAI.Computability.DegreeRigidity.Constructibility.RelativePersistentExtension | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open TransitiveNameModel BoundedSetTheory ElementaryModel SetDegreeDecoding PersistentRestrictions | |
| universe u | |
| theorem persistent_graph_construction (M : ZFSet.{u}) [Countable (Conditions M)] | |
| (hM : Transitive M) (hT : SourceT M) | |
| (ρ : modelIdeal M hM hT ≃o modelIdeal M hM hT) | |
| (hρ : Persistent (modelIdeal M hM hT) ρ) : | |
| ∃ t : Construction.{u}, ∃ P : Oracle, P ∈ modelReals M ∧ | |
| t.Indexed M ∧ t.Certified (groundReals M) P ∧ | |
| t.value (groundReals M) P = automorphismSet ρ := | |
| construction_of_relativeModel M hM hT _ | |
| (persistent_modelIdeal_graph_mem_relativeModel M hM hT ρ hρ) | |
| theorem persistent_extension_construction (M : ZFSet.{u}) [Countable (Conditions M)] | |
| (hM : Transitive M) (hT : SourceT M) | |
| (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) : | |
| ∃ σ : modelIdeal M hM hT ≃o modelIdeal M hM hT, | |
| Extends hIM ρ σ ∧ Persistent (modelIdeal M hM hT) σ ∧ | |
| ∃ t : Construction.{u}, ∃ P : Oracle, P ∈ modelReals M ∧ | |
| t.Indexed M ∧ t.Certified (groundReals M) P ∧ | |
| t.value (groundReals M) P = automorphismSet σ := by | |
| obtain ⟨σ,he,hσ,hm⟩ := persistent_extension_in_relativeModel M hM hT I ρ hρ hz hIM | |
| exact ⟨σ,he,hσ,construction_of_relativeModel M hM hT _ hm⟩ | |
| theorem persistent_graph_definition (M : ZFSet.{u}) [Countable (Conditions M)] | |
| (hM : Transitive M) (hT : SourceT M) | |
| (ρ : modelIdeal M hM hT ≃o modelIdeal M hM hT) | |
| (hρ : Persistent (modelIdeal M hM hT) ρ) : | |
| ∃ (o : Ordinal.{u}) (n : ℕ) (p : SentenceForm) | |
| (c : Fin n → Construction.{u}) (P : Oracle), | |
| o.toZFSet ∈ M ∧ P ∈ modelReals M ∧ | |
| (Construction.define o n p c).Indexed M ∧ | |
| (Construction.define o n p c).Certified (groundReals M) P ∧ | |
| satisfactionSet (level (groundReals M) o) ∈ M ∧ | |
| ∀ z, z ∈ automorphismSet ρ ↔ z ∈ level (groundReals M) o ∧ | |
| p.Sat (level (groundReals M) o : Set ZFSet) | |
| (cons z (tupleEnv (fun i => (c i).value (groundReals M) (realPart P i)))) := | |
| construction_definition_of_relativeModel M hM hT _ | |
| (persistent_modelIdeal_graph_mem_relativeModel M hM hT ρ hρ) | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |