openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /FixedRelativeGround.lean
| import OAI.Computability.DegreeRigidity.SetModels.NoNewOrdinals | |
| import OAI.Computability.DegreeRigidity.Constructibility.RelativeModelComparison | |
| import OAI.Computability.DegreeRigidity.Representation.SourceTGenericPersistence | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open TransitiveNameModel BoundedSetTheory CountableForcing | |
| universe u | |
| theorem relativeModel_generic_fixed_parameter (M : ZFSet.{u}) | |
| (hM : Transitive M) (hT : SourceT M) {R c o : ZFSet.{u}} | |
| [Preorder (Conditions c)] [OrderTop (Conditions c)] (hR : R ∈ M) | |
| (hc : c ∈ M) (ho : o ∈ M) | |
| (hos : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s) | |
| (G : GenericFilter (Conditions c)) (hG : AtomicForcing.GroundGeneric M G) : | |
| relativeModel (genericExtensionSet M c G.carrier) R = relativeModel M R := by | |
| obtain ⟨p,hp⟩ := G.nonempty | |
| have ht := G.upper le_top hp | |
| 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 ht | |
| apply relativeModel_eq_of_same_ordinals _ M R | |
| (genericExtensionSet_transitive M c hM G.carrier) | |
| (extension_sourceT M hM hT hc ho hos G hG ht) (hME hR) hM hT hR | |
| intro δ | |
| exact ⟨extension_ordinal_ground M c hM hT G.carrier δ,fun hδ => hME hδ⟩ | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |