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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 47 | 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
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