File size: 3,796 Bytes
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 63 64 65 66 67 68 | import OAI.Computability.DegreeRigidity.Syntax.DefinedPrefixRemoval
import OAI.Computability.DegreeRigidity.Representation.OwnSourceEquation
namespace OAI
namespace TuringRigidity.RelativeConstructible
open TransitiveNameModel BoundedSetTheory CountableForcing AtomicForcing RecursiveNames
open CohenColumnRealName ElementaryModel PersistentRestrictions SetDegreeDecoding BoundedForcing
open GenericIdentity
attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder
def ColumnRealizesExtension (M K a E : ZFSet.{0}) (A : Oracle) : Prop :=
∃ H : GenericFilter (Conditions (poset K)), GroundGeneric M H ∧
genericExtensionSet M (poset K) H.carrier = E ∧
(realName K a).val H.carrier = realCode A
theorem own_program_in_common_extension (M K a : ZFSet.{0}) (δ : Ordinal.{0})
(Z : ZFSet.{0}) (φ : SentenceForm) (f : Name (Conditions (poset K)))
(hfv : ∀ H : GenericFilter (Conditions (poset K)), GroundGeneric M H →
f.val H.carrier = definedSubset
(level (groundReals (genericExtensionSet M (poset K) H.carrier)) δ)
φ (fun _ : Fin φ.bound => Z))
(p : OracleCode) (R : Oracle)
(hprog : ∀ H : GenericFilter (Conditions (poset K)), GroundGeneric M H →
∀ A : Oracle, (realName K a).val H.carrier = realCode A →
OwnProgramAt (genericExtensionSet M (poset K) H.carrier) (f.val H.carrier) p R A)
(G : GenericFilter (Conditions (poset K))) (hG : GroundGeneric M G)
(A : Oracle) (hA : ColumnRealizesExtension M K a
(genericExtensionSet M (poset K) G.carrier) A) :
OwnProgramAt (genericExtensionSet M (poset K) G.carrier) (f.val G.carrier) p R A := by
obtain ⟨H,hH,he,hv⟩ := hA
have hf := defined_value_extension_eq M (poset K) δ Z φ f hfv H G hH hG he
have h := hprog H hH A hv
rwa [he,hf] at h
theorem sourceEquation_of_common_extension (M K a : ZFSet.{0})
(δ : Ordinal.{0}) (Z : ZFSet.{0}) (φ : SentenceForm) (f : Name (Conditions (poset K)))
(hfv : ∀ H : GenericFilter (Conditions (poset K)), GroundGeneric M H →
f.val H.carrier = definedSubset
(level (groundReals (genericExtensionSet M (poset K) H.carrier)) δ)
φ (fun _ : Fin φ.bound => Z))
(p : OracleCode) (P : Oracle)
(hprog : ∀ H : GenericFilter (Conditions (poset K)), GroundGeneric M H →
∀ A : Oracle, (realName K a).val H.carrier = realCode A →
OwnProgramAt (genericExtensionSet M (poset K) H.carrier) (f.val H.carrier) p P A)
(G : GenericFilter (Conditions (poset K))) (hG : GroundGeneric M G)
(hE : Transitive (genericExtensionSet M (poset K) G.carrier))
(hTE : SourceT (genericExtensionSet M (poset K) G.carrier))
[Countable (Conditions (genericExtensionSet M (poset K) G.carrier))]
(I : CountableIdeal) (ρ : I ≃o I)
(hOwn : OwnDegreeExtension (genericExtensionSet M (poset K) G.carrier)
(idealSet I) (automorphismSet ρ) (f.val G.carrier))
(Y L R : Oracle) (hL : GenericCoding.InfiniteOdd L) (hR : GenericCoding.InfiniteOdd R)
(hideal : PrincipalIntersection Y (join Y L) (join Y R))
(hlifts : ∀ A ∈ ({Y,L,R,GenericCoding.code Y L,GenericCoding.code Y R} : Set Oracle),
ColumnRealizesExtension M K a (genericExtensionSet M (poset K) G.carrier) A) :
SourceEquation p P (Y,L,R) := by
have hrep (A : Oracle) (hA : A ∈ ({Y,L,R,GenericCoding.code Y L,GenericCoding.code Y R} : Set Oracle)) :=
own_program_in_common_extension M K a δ Z φ f hfv p P hprog G hG A (hlifts A hA)
exact sourceEquation_of_own_extension _ hE hTE I ρ _ hOwn p P Y L R hL hR hideal
(hrep Y (by simp)) (hrep L (by simp)) (hrep R (by simp))
(hrep (GenericCoding.code Y L) (by simp)) (hrep (GenericCoding.code Y R) (by simp))
end TuringRigidity.RelativeConstructible
end OAI
|