openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Representation /GenericAssembly.lean
| import OAI.Computability.DegreeRigidity.SetModels.IdealSelection | |
| namespace OAI | |
| namespace TuringRigidity | |
| theorem representative_from_generic_coding (π : Degree ≃o Degree) | |
| (F : Oracle → Oracle) (hF : Measurable F) | |
| (C : Oracle → Oracle → Oracle) (hC : Measurable (Function.uncurry C)) | |
| (G : Bool → Oracle → Oracle) (hG : ∀ b, Measurable (G b)) | |
| (hgen : ∀ A b, degree (F (G b A)) = π (degree (G b A))) | |
| (hcoded : ∀ A b, degree (F (C A (G b A))) = π (degree (C A (G b A)))) | |
| (hideal : ∀ A a, a ≤ degree A ↔ | |
| a ≤ degree (join (C A (G false A)) (G false A)) ∧ | |
| a ≤ degree (join (C A (G true A)) (G true A))) : | |
| ∃ H : Oracle → Oracle, Measurable H ∧ ∀ A, degree (H A) = π (degree A) := by | |
| let L : Bool → Oracle → Oracle := fun b A => join (F (C A (G b A))) (F (G b A)) | |
| have hL (b : Bool) : Measurable (L b) := | |
| join_measurable _ _ (hF.comp (hC.comp (measurable_id.prodMk (hG b)))) (hF.comp (hG b)) | |
| have hdegree (A : Oracle) (b : Bool) : | |
| degree (L b A) = π (degree (join (C A (G b A)) (G b A))) := by | |
| dsimp only [L] | |
| change degree (F (C A (G b A))) ⊔ degree (F (G b A)) = | |
| π (degree (C A (G b A)) ⊔ degree (G b A)) | |
| rw [hcoded A b, hgen A b, π.map_sup] | |
| apply borel_of_ideal_intersection (L false) (L true) (hL false) (hL true) | |
| (fun A => π (degree A)) | |
| intro A a | |
| rw [hdegree A false,hdegree A true] | |
| exact common_lower_transport π _ _ _ (hideal A) a | |
| end TuringRigidity | |
| end OAI | |