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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 | import OAI.Computability.DegreeRigidity.Representation.GenericTruth
import OAI.Computability.DegreeRigidity.Representation.GenericFiberIdeal
import OAI.Computability.DegreeRigidity.Representation.SourceIdentity
namespace OAI
namespace TuringRigidity.GenericSourceTransfer
open FiniteShuffle ShuffleRequirements PairGenericSelection GenericFactor GenericIdentity Set
@[simp] theorem triple_join (Y Z : Oracle) :
GenericTruth.triple (join Y Z) = (Y,column false Z,column true Z) := by
have hl : column false (join Y Z) = Y := by funext n; simp [column]
have hr : column true (join Y Z) = Z := by funext n; simp [column]
simp [GenericTruth.triple,hl,hr]
theorem selected_borel_truth (p : OracleCode) (P : Oracle)
(E : ℕ → List Bool → Prop) (hE : ∀ n, DenseOpen (E n)) :
∃ U : Set Oracle, IsOpen U ∧
∃ F : ℕ → List Bool → Prop, (∀ n, DenseOpen (F n)) ∧
∃ Z : {Y : Oracle // GenericFor F Y} → Oracle, Measurable Z ∧
(∀ Y b, GenericFor E (column b (Z Y)) ∧ GenericCoding.InfiniteOdd (column b (Z Y)) ∧
∀ X, GenericFor E (GenericCoding.code X (column b (Z Y)))) ∧
(∀ Y a, a ≤ degree Y.val ↔
a ≤ degree (join Y.val (column false (Z Y))) ∧
a ≤ degree (join Y.val (column true (Z Y)))) ∧
∀ Y, SourceEquation p P (Y.val,column false (Z Y),column true (Z Y)) ↔
join Y.val (Z Y) ∈ U := by
obtain ⟨U,hU,D,hD,htruth⟩ := GenericTruth.sourceEquation_generic_truth p P
obtain ⟨hF,Z,hZ,hjoint,hgen,hI⟩ := borel_joint_ideal_selection D E hD hE
refine ⟨U,hU,FirstFamily D,hF,Z,hZ,hgen,hI,?_⟩
intro Y
simpa only [triple_join] using htruth (join Y.val (Z Y)) (hjoint Y)
theorem recover_at (π : Degree ≃o Degree) (p : OracleCode) (P Y G0 G1 : Oracle)
(hG0 : RepresentsAt π p P G0) (hG1 : RepresentsAt π p P G1)
(hC0 : RepresentsAt π p P (GenericCoding.code Y G0))
(hC1 : RepresentsAt π p P (GenericCoding.code Y G1))
(hi0 : GenericCoding.InfiniteOdd G0) (hi1 : GenericCoding.InfiniteOdd G1)
(hI : PrincipalIntersection Y (join Y G0) (join Y G1))
(he : SourceEquation p P (Y,G0,G1)) : RepresentsAt π p P Y :=
(source_5_5_iff π p P Y G0 G1 hG0 hG1 hC0 hC1 hi0 hi1 hI).mp he
end TuringRigidity.GenericSourceTransfer
end OAI
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