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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 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 | 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
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