openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Representation /CommonExtensionProgram.lean
| 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 | |