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