openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Representation /GenericRealColumnRealization.lean
| import OAI.Computability.DegreeRigidity.Effective.PrefixSingletonRealization | |
| import OAI.Computability.DegreeRigidity.CohenForcing.CohenPartitionRealization | |
| import OAI.Computability.DegreeRigidity.CohenForcing.CohenRemainderTransport | |
| import OAI.Computability.DegreeRigidity.Representation.OriginalRealCohenFactorization | |
| import OAI.Computability.DegreeRigidity.Representation.CommonExtensionProgram | |
| import OAI.Computability.DegreeRigidity.SetModels.InternalShuffleGeneric | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open TransitiveNameModel BoundedSetTheory CountableForcing InternalCountableOrdinals | |
| open CohenColumnRealName | |
| attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder | |
| theorem generic_real_column_realizes (M K a : ZFSet.{0}) | |
| (hM : Transitive M) (hT : SourceT M) (hK : FirstUncountable M K) (ha : a ∈ K) | |
| (G : GenericFilter (Conditions (poset K))) (hG : AtomicForcing.GroundGeneric M G) | |
| (A : Oracle) (hAE : realCode A ∈ genericExtensionSet M (poset K) G.carrier) | |
| (hA : AtomicForcing.GroundGeneric M (InternalCohen.pushFilter (CohenBorelForcing.realFilter A))) : | |
| ColumnRealizesExtension M K a (genericExtensionSet M (poset K) G.carrier) A := by | |
| obtain ⟨hN,L,hL,hLe⟩ := OriginalRealCohenFactorization.real_factorization M K hM hT hK | |
| G hG (realCode A) hAE (realCode_subset A) | |
| have haM := hM K hK.2.1 a ha | |
| have hsingle := singleton_mem M hM hT.pairing haM | |
| have hsK : ({a} : ZFSet.{0}) ⊆ K := by | |
| intro x hx | |
| exact ZFSet.mem_singleton.mp hx ▸ ha | |
| obtain ⟨R,hR,hRe⟩ := CohenRemainderTransport.transport_to_remainder M _ K {a} | |
| hM hT hN.1 hN.2.1 hN.2.2.1 hK hsingle hsK | |
| (Or.inr (InternalCountableClosure.singleton_countable M a hM hT haM)) L hL | |
| obtain ⟨J,hJ,hJe,hJA⟩ := PrefixSingletonRealization.singleton_realization M a hM hT haM A hA | |
| have hRJ : AtomicForcing.GroundGeneric (genericExtensionSet M (poset {a}) J.carrier) R := by | |
| rw [hJe] | |
| exact hR | |
| obtain ⟨U,hU,hUe,hUA⟩ := CohenPartitionRealization.reassemble_column M K a hM hT hK.2.1 ha A | |
| J hJ hJA R hRJ | |
| refine ⟨U,hU,hUe.trans ?_,hUA⟩ | |
| rw [hJe] | |
| exact hRe.trans hLe | |
| theorem shuffled_real_column_realizes (M K a : ZFSet.{0}) | |
| (hM : Transitive M) (hT : SourceT M) (hK : FirstUncountable M K) (ha : a ∈ K) | |
| (G : GenericFilter (Conditions (poset K))) (hG : AtomicForcing.GroundGeneric M G) | |
| (Y L : Oracle) | |
| (hYE : realCode Y ∈ genericExtensionSet M (poset K) G.carrier) | |
| (hLE : realCode L ∈ genericExtensionSet M (poset K) G.carrier) | |
| (hL : AtomicForcing.GroundGeneric M (InternalCohen.pushFilter (CohenBorelForcing.realFilter L))) : | |
| ColumnRealizesExtension M K a (genericExtensionSet M (poset K) G.carrier) (GenericCoding.code Y L) := by | |
| obtain ⟨hE,hTE,_,_⟩ := RegularTreeExtension.extension_properties M (poset K) hM hT | |
| (CohenGroundPoset.conditions_mem M _ hM hT | |
| (product_mem M hM hT.pairing hT.union hT.powerSet hT.separation.finitePrefix.bounded | |
| hK.2.1 (sourceT_omega_mem M hM hT))) G hG | |
| have hcode := sourceT_real_lower _ hE hTE (sourceT_real_join _ hE hTE hYE hLE) | |
| (GenericCoding.code_reduces Y L) | |
| exact generic_real_column_realizes M K a hM hT hK ha G hG _ hcode | |
| (InternalCohen.groundGeneric_shuffle M hM hT L hL Y) | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |