| import OAI.Computability.DegreeRigidity.CohenForcing.CohenPrefixOverwrite |
| import OAI.Computability.DegreeRigidity.Effective.UniformTableSimulation |
| import OAI.Computability.DegreeRigidity.SetModels.ModelDegreeUniverse |
| import OAI.Computability.DegreeRigidity.Constructibility.ConstructibleGroundReals |
|
|
| namespace OAI |
|
|
|
|
| namespace TuringRigidity.RelativeConstructible |
| open TransitiveNameModel BoundedSetTheory CountableForcing RecursiveNames AtomicForcing |
| open CohenGroundPoset CohenColumnRealName CohenFiniteFlip FiniteOverwrite |
| open TableIndices IndexMatrix |
| open ElementaryModel |
| attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder |
| CohenNiceNameConstruction.cohenTop |
|
|
| theorem equidegree_mem_ownDegree (E u : ZFSet.{0}) (hE : Transitive E) (hT : SourceT E) |
| (hu : u ∈ degreeUniverse (groundReals E)) (A B : Oracle) |
| (hB : realCode B ∈ u) (he : degree A = degree B) : realCode A ∈ u := by |
| obtain ⟨C,hC,rfl⟩ := (mem_degreeUniverse _ _).mp hu |
| have hmem := actual_degreeCode_membership E hE hT (groundReals E) |
| (realCode_mem_groundReals E) ((realCode_mem_groundReals E C).mp hC) |
| exact (hmem A).mpr (he.trans ((hmem B).mp hB)) |
|
|
| theorem defined_value_extension_eq (M c : ZFSet.{0}) (δ : Ordinal.{0}) |
| (Z : ZFSet.{0}) (φ : SentenceForm) (f : Name (Conditions c)) |
| (hfv : ∀ H : GenericFilter (Conditions c), GroundGeneric M H → |
| f.val H.carrier = definedSubset (level (groundReals (genericExtensionSet M c H.carrier)) δ) |
| φ (fun _ : Fin φ.bound => Z)) |
| (G H : GenericFilter (Conditions c)) (hG : GroundGeneric M G) (hH : GroundGeneric M H) |
| (he : genericExtensionSet M c G.carrier = genericExtensionSet M c H.carrier) : |
| f.val G.carrier = f.val H.carrier := by |
| rw [hfv G hG,hfv H hH,he] |
|
|
| theorem remove_defined_program_prefix (M K a : ZFSet.{0}) |
| (hM : Transitive M) (hT : SourceT M) (hK : K ∈ M) (ha : a ∈ K) |
| (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)) |
| (R A₀ : Oracle) (d ℓ : ℕ) |
| (hprog : ∀ H : GenericFilter (Conditions (poset K)), GroundGeneric M H → |
| ∀ A : Oracle, (realName K a).val H.carrier = realCode A → |
| (∀ n < ℓ, A n = A₀ n) → ∃ X : Oracle, |
| Represents (join A R) (machine d) X ∧ |
| ∃ u ∈ degreeUniverse (groundReals (genericExtensionSet M (poset K) H.carrier)), |
| ∃ v ∈ degreeUniverse (groundReals (genericExtensionSet M (poset K) H.carrier)), |
| realCode A ∈ u ∧ realCode X ∈ v ∧ ZFSet.pair u v ∈ f.val H.carrier) : |
| ∃ e : ℕ, ∀ H : GenericFilter (Conditions (poset K)), GroundGeneric M H → |
| ∀ A : Oracle, (realName K a).val H.carrier = realCode A → ∃ X : Oracle, |
| Represents (join A R) (machine e) X ∧ |
| ∃ u ∈ degreeUniverse (groundReals (genericExtensionSet M (poset K) H.carrier)), |
| ∃ v ∈ degreeUniverse (groundReals (genericExtensionSet M (poset K) H.carrier)), |
| realCode A ∈ u ∧ realCode X ∈ v ∧ ZFSet.pair u v ∈ f.val H.carrier := by |
| let s := FiniteShuffle.initial A₀ ℓ |
| obtain ⟨e,he⟩ := (overwrite_join_program s R).table_simulation d |
| refine ⟨e,fun H hH A hv => ?_⟩ |
| let Hs := prefixFilter K a s A H |
| obtain ⟨hHs,hExt,hval⟩ := prefixFilter_properties M K a hM hT hK ha s A H hH hv |
| have hprefix : ∀ n < ℓ, overwrite s A n = A₀ n := by |
| intro n hn |
| rw [overwrite_below _ _ _ (by simpa [s] using hn),FiniteShuffle.prefix_getD _ _ _ hn] |
| obtain ⟨X,hX,u,hu,v,hv',hAu,hXv,hpair⟩ := hprog Hs hHs (overwrite s A) hval hprefix |
| have hfEq := defined_value_extension_eq M (poset K) δ Z φ f hfv Hs H hHs hH hExt |
| rw [hExt] at hu hv' |
| rw [hfEq] at hpair |
| obtain ⟨hE,hTE,_,_⟩ := RegularTreeExtension.extension_properties M (poset K) hM hT |
| (manyColumn_internal M K hM hT hK).1 H hH |
| exact ⟨X,he A X hX,u,hu,v,hv', |
| equidegree_mem_ownDegree _ u hE hTE hu A (overwrite s A) hAu |
| (degree_overwrite s A).symm,hXv,hpair⟩ |
| |
| end TuringRigidity.RelativeConstructible |
| |
| end OAI |
| |
