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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 | 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
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