| import OAI.Computability.DegreeRigidity.CohenForcing.CohenPrefixHull |
| import OAI.Computability.DegreeRigidity.Representation.GroundPrefixOverwrite |
|
|
| namespace OAI |
|
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|
|
| namespace TuringRigidity.FiniteOverwriteHull |
| open TransitiveNameModel BoundedSetTheory RealGeneratedModel FiniteOverwrite |
| open CountableForcing InternalCohen CohenBorelForcing |
| attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder |
|
|
| theorem contains_iff_of_degree (M N : ZFSet.{0}) (A B : Oracle) |
| (he : degree A = degree B) : Contains M (realCode A) N ↔ Contains M (realCode B) N := by |
| obtain ⟨hAB,hBA⟩ := (degree_eq_iff A B).mp he |
| constructor |
| · rintro ⟨hN,hT,hMN,hA⟩ |
| exact ⟨hN,hT,hMN,sourceT_real_lower N hN hT hA hBA⟩ |
| · rintro ⟨hN,hT,hMN,hB⟩ |
| exact ⟨hN,hT,hMN,sourceT_real_lower N hN hT hB hAB⟩ |
|
|
| theorem hull_eq_of_degree (M : ZFSet.{0}) (A B : Oracle) |
| (he : degree A = degree B) : RealGeneratedModel.hull M (realCode A) = RealGeneratedModel.hull M (realCode B) := by |
| classical |
| have hc := fun N => contains_iff_of_degree M N A B he |
| by_cases hA : ∃ N, Contains M (realCode A) N |
| · have hB : ∃ N, Contains M (realCode B) N := |
| ⟨hA.choose,(hc _).mp hA.choose_spec⟩ |
| apply ZFSet.ext |
| intro z |
| rw [mem_hull M _ z hA,mem_hull M _ z hB] |
| exact ⟨fun h N hN => h N ((hc N).mpr hN),fun h N hN => h N ((hc N).mp hN)⟩ |
| · have hB : ¬ ∃ N, Contains M (realCode B) N := fun h => |
| hA ⟨h.choose,(hc _).mpr h.choose_spec⟩ |
| simp only [RealGeneratedModel.hull,dite_eq_right hA,dite_eq_right hB] |
|
|
| theorem overwrite_hull (M : ZFSet.{0}) (s : List Bool) (A : Oracle) : |
| RealGeneratedModel.hull M (realCode (overwrite s A)) = RealGeneratedModel.hull M (realCode A) := |
| hull_eq_of_degree M _ _ (degree_overwrite s A) |
|
|
| theorem overwrite_extension (M : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M) |
| (A : Oracle) (hA : AtomicForcing.GroundGeneric M (pushFilter (realFilter A))) |
| (s : List Bool) : |
| genericExtensionSet M conditions (pushFilter (realFilter (overwrite s A))).carrier = |
| genericExtensionSet M conditions (pushFilter (realFilter A)).carrier := by |
| exact prefix_extension_eq_of_hull M hM hT _ _ |
| (GroundPrefixOverwrite.overwrite_generic M hM hT A hA s) hA (overwrite_hull M s A) |
|
|
| end TuringRigidity.FiniteOverwriteHull |
|
|
| end OAI |
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|
