openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /CountedOriginalRealQuotient.lean
| import OAI.Computability.DegreeRigidity.Representation.OriginalRealQuotientModel | |
| import OAI.Computability.DegreeRigidity.CohenForcing.CountedCohenConditions | |
| import OAI.Computability.DegreeRigidity.SetModels.InternalQuotientCountability | |
| namespace OAI | |
| namespace TuringRigidity.CountedOriginalRealQuotient | |
| open TransitiveNameModel BoundedSetTheory CountableForcing RecursiveNames | |
| open CohenGroundPoset InternalRegularOperations InternalRegularAlgebra InternalBooleanSyntax | |
| open InternalProjectedGeneric | |
| attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder | |
| CohenNiceNameConstruction.cohenTop | |
| attribute [local instance] codeOrder codePreorder | |
| theorem original_name_counted_quotient (M K : ZFSet.{0}) | |
| (hM : Transitive M) (hT : SourceT M) (hK : K ∈ M) | |
| (τ : Name (Conditions (conditions (ZFSet.prod K ZFSet.omega)))) | |
| (hτ : τ.encode (label (conditions (ZFSet.prod K ZFSet.omega))) ∈ M) : | |
| ∃ C ∈ M, ∃ _hCK : C ⊆ K, (C = ∅ ∨ InternallyCountable M C) ∧ | |
| ∃ E : ℕ → ZFSet.{0}, orbitGraph E ∈ M ∧ | |
| (∀ n, E n ∈ M ∧ E n ⊆ conditions (ZFSet.prod C ZFSet.omega)) ∧ | |
| ∃ B ∈ M, ∃ Q ∈ M, ∃ A ∈ M, | |
| (∀ U, U ∈ B ↔ U ∈ M ∧ IsCode (conditions (ZFSet.prod C ZFSet.omega)) U) ∧ | |
| (∀ F, F ∈ Q ↔ F ∈ M ∧ F ⊆ B) ∧ | |
| A = InternalGeneratedAlgebra.generated (conditions (ZFSet.prod C ZFSet.omega)) B Q | |
| (InternalBooleanBits.seeds (conditions (ZFSet.prod C ZFSet.omega)) B E) ∧ | |
| Closed (conditions (ZFSet.prod C ZFSet.omega)) B Q A ∧ | |
| ∃ σ : Name (Conditions (positive A)), σ.encode (label (positive A)) ∈ M ∧ | |
| ∃ hBA : ZFSet.prod C ZFSet.omega ⊆ ZFSet.prod K ZFSet.omega, | |
| ∀ G : GenericFilter (Conditions (conditions (ZFSet.prod K ZFSet.omega))), | |
| AtomicForcing.GroundGeneric M G → τ.val G.carrier ⊆ ZFSet.omega → | |
| let J := InternalCohenProjectedGeneric.projected _ _ hBA G | |
| let X := τ.val G.carrier | |
| ∃ H : GenericFilter (Conditions (positive A)), AtomicForcing.GroundGeneric M H ∧ | |
| σ.val H.carrier = X ∧ | |
| genericExtensionSet M (positive A) H.carrier = RealGeneratedModel.hull M X ∧ | |
| RealGeneratedModel.Contains M X (RealGeneratedModel.hull M X) ∧ | |
| let q := InternalQuotientConditions.conditions (conditions (ZFSet.prod C ZFSet.omega)) A | |
| (genericFilterSet (positive A) H.carrier) | |
| q ∈ RealGeneratedModel.hull M X ∧ InternalCollapse.orderSet q ∈ RealGeneratedModel.hull M X ∧ | |
| InternallyCountable (RealGeneratedModel.hull M X) q ∧ | |
| ∃ Gq : GenericFilter (Conditions q), AtomicForcing.GroundGeneric (RealGeneratedModel.hull M X) Gq ∧ | |
| genericExtensionSet (RealGeneratedModel.hull M X) q Gq.carrier = | |
| genericExtensionSet M (conditions (ZFSet.prod C ZFSet.omega)) J.carrier ∧ | |
| genericFilterSet q Gq.carrier = | |
| genericFilterSet (conditions (ZFSet.prod C ZFSet.omega)) J.carrier ∧ | |
| genericExtensionSet (RealGeneratedModel.hull M X) q Gq.carrier ⊆ | |
| genericExtensionSet M (conditions (ZFSet.prod K ZFSet.omega)) G.carrier ∧ | |
| ∀ p ∈ G.carrier, ∃ r ∈ Gq.carrier, | |
| label q r = label _ (InternalCohenFactor.project _ _ hBA p) := by | |
| obtain ⟨C,hC,hCK,hct,E,hgraph,hE,B,hBM,Q,hQM,A,hAM,hB,hQ,hAe,hA,σ,hσ,hBA,hall⟩ := | |
| OriginalRealQuotientModel.original_name_quotient M K hM hT hK τ hτ | |
| have hcM := (manyColumn_internal M C hM hT hC).1 | |
| have hcount := CountedCohenConditions.many_column_conditions_countable M C hM hT hC hct | |
| refine ⟨C,hC,hCK,hct,E,hgraph,hE,B,hBM,Q,hQM,A,hAM,hB,hQ,hAe,hA,σ,hσ,hBA,?_⟩ | |
| intro G hG hreal | |
| obtain ⟨H,hHG,hval,hHull,hprop,hq,ho,Gq,hGq,hext,hcode,hsub,hcond⟩ := hall G hG hreal | |
| refine ⟨H,hHG,hval,hHull,hprop,hq,ho,?_,Gq,hGq,hext,hcode,hsub,hcond⟩ | |
| apply InternalCountableClosure.countable_subset _ _ _ hprop.1 hprop.2.1 | |
| (hprop.2.2.1 hcM) hq (InternalQuotientConditions.conditions_subset _ A _) | |
| (InternalQuotientCountability.countability_upward M _ _ hprop.2.2.1 hcount) | |
| obtain ⟨p,_⟩ := Gq.nonempty | |
| exact ⟨label _ p,label_mem _ p⟩ | |
| end TuringRigidity.CountedOriginalRealQuotient | |
| end OAI | |