File size: 4,287 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 69 70 71 72 73 74 75 76 77 78 | 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
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