openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Representation /GenericCodingBorel.lean
| import OAI.Computability.DegreeRigidity.Representation.GenericCoding | |
| namespace OAI | |
| namespace TuringRigidity.GenericCoding | |
| open Set UniformOracle BinarySeries | |
| theorem query_measurable {X : Type*} [MeasurableSpace X] | |
| (A : X → Oracle) (q : X → ℕ) (hA : Measurable A) (hq : Measurable q) : | |
| Measurable (fun x => A x (q x)) := by | |
| apply measurable_to_countable' | |
| intro b | |
| change MeasurableSet {x | A x (q x) = b} | |
| have he : {x | A x (q x) = b} = ⋃ n : ℕ, {x | q x = n} ∩ {x | A x n = b} := by | |
| ext x | |
| simp only [mem_ofPred_eq,mem_iUnion,mem_inter_iff] | |
| exact ⟨fun h => ⟨q x,rfl,h⟩,fun ⟨n,hn,h⟩ => by simpa [hn] using h⟩ | |
| rw [he] | |
| exact MeasurableSet.iUnion (fun n => (measurableSet_eq_fun hq measurable_const).inter | |
| (measurableSet_eq_fun ((measurable_pi_apply n).comp hA) measurable_const)) | |
| theorem count_measurable (n : ℕ) : Measurable (fun G : Oracle => count G n) := by | |
| induction n with | |
| | zero => exact measurable_const | |
| | succ n ih => | |
| have hb : Measurable (fun G : Oracle => bit G (2*n+1)) := | |
| (measurable_of_countable (fun b : Bool => if b then (1 : ℕ) else 0)).comp | |
| (measurable_pi_apply (2*n+1)) | |
| exact (measurable_of_countable (fun z : ℕ × ℕ => z.1+z.2)).comp (ih.prodMk hb) | |
| theorem code_measurable : Measurable (Function.uncurry code) := by | |
| apply Measurable.of_eval | |
| intro n | |
| have hc : Measurable (fun p : Oracle × Oracle => count p.2 n) := (count_measurable n).comp measurable_snd | |
| have ha := query_measurable (fun p : Oracle × Oracle => p.1) (fun p => count p.2 n) measurable_fst hc | |
| have hi : Measurable (fun p : Oracle × Oracle => 2*(n-count p.2 n)) := | |
| (measurable_of_countable (fun k : ℕ => 2*(n-k))).comp hc | |
| have he := query_measurable (fun p : Oracle × Oracle => p.2) _ measurable_snd hi | |
| have ho : Measurable (fun p : Oracle × Oracle => p.2 (2*n+1)) := | |
| (measurable_pi_apply (2*n+1)).comp measurable_snd | |
| exact (measurable_of_countable (fun z : Bool × Bool × Bool => if z.1 then z.2.1 else z.2.2)).comp | |
| (ho.prodMk (ha.prodMk he)) | |
| end TuringRigidity.GenericCoding | |
| end OAI | |