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Learning.measurable_uncurry_pullCount๐Ÿ”—

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measurable_uncurry_pullCount๐Ÿ”—

LemmaLearning.measurable_uncurry_pullCount
Details

No docstring.

๐Ÿ”—theorem
Learning.measurable_uncurry_pullCount.{u_1, u_3} {๐“ : Type u_1}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“} {mฮฉ : MeasurableSpace ฮฉ}
  [DecidableEq ๐“] {A : โ„• โ†’ ฮฉ โ†’ ๐“} [MeasurableEq ๐“]
  (hA : โˆ€ (n : โ„•), Measurable (A n)) (t : โ„•) :
  Measurable fun p => pullCount A (Prod.snd p) t (Prod.fst p)
Learning.measurable_uncurry_pullCount.{u_1,
    u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  {m๐“ : MeasurableSpace ๐“}
  {mฮฉ : MeasurableSpace ฮฉ} [DecidableEq ๐“]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} [MeasurableEq ๐“]
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (t : โ„•) :
  Measurable fun p =>
    pullCount A (Prod.snd p) t
      (Prod.fst p)

Code

lemma measurable_uncurry_pullCount [MeasurableEq ๐“]
    (hA : โˆ€ n, Measurable (A n)) (t : โ„•) :
    Measurable (fun p : ฮฉ ร— ๐“ โ†ฆ pullCount A p.2 t p.1)
Type uses (1)
Body uses (1)
Used by (5)

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Proof
by
  simp_rw [pullCount_eq_sum]
  have h_meas s : Measurable (fun h : ฮฉ ร— ๐“ โ†ฆ if A s h.1 = h.2 then 1 else 0) := by
    refine Measurable.ite ?_ (by fun_prop) (by fun_prop)
    exact measurableSet_eq_fun (by fun_prop) (by fun_prop)
  fun_prop

Dependency graph

Type dependencies (1)

pullCount๐Ÿ”—

DefinitionLearning.pullCount
Details

Number of times action a was chosen up to time t (excluding t).

๐Ÿ”—def
Learning.pullCount.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„•
Learning.pullCount.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“)
  (t : โ„•) (ฯ‰ : ฮฉ) : โ„•

Code

noncomputable
def pullCount (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„• :=
  #(filter (fun s โ†ฆ A s ฯ‰ = a) (range t))
Used by (146)

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