LeanMachineLearning exposition

Learning.measurable_uncurry_sumRewards_comp๐Ÿ”—

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measurable_uncurry_sumRewards_comp๐Ÿ”—

LemmaLearning.measurable_uncurry_sumRewards_comp
Details

No docstring.

๐Ÿ”—theorem
Learning.measurable_uncurry_sumRewards_comp.{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“} [Countable ๐“]
  [MeasurableSingletonClass ๐“] {R' : โ„• โ†’ ฮฉ โ†’ โ„}
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hR' : โˆ€ (n : โ„•), Measurable (R' n)) {f : ฮฉ โ†’ ๐“} (hf : Measurable f)
  {g : ฮฉ โ†’ โ„•} (hg : Measurable g) :
  Measurable fun ฯ‰ => sumRewards A R' (f ฯ‰) (g ฯ‰) ฯ‰
Learning.measurable_uncurry_sumRewards_comp.{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“} [Countable ๐“]
  [MeasurableSingletonClass ๐“]
  {R' : โ„• โ†’ ฮฉ โ†’ โ„}
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hR' : โˆ€ (n : โ„•), Measurable (R' n))
  {f : ฮฉ โ†’ ๐“} (hf : Measurable f)
  {g : ฮฉ โ†’ โ„•} (hg : Measurable g) :
  Measurable fun ฯ‰ =>
    sumRewards A R' (f ฯ‰) (g ฯ‰) ฯ‰

Code

lemma measurable_uncurry_sumRewards_comp [Countable ๐“] [MeasurableSingletonClass ๐“]
    {R' : โ„• โ†’ ฮฉ โ†’ โ„} (hA : โˆ€ n, Measurable (A n)) (hR' : โˆ€ n, Measurable (R' n)) {f : ฮฉ โ†’ ๐“}
    (hf : Measurable f) {g : ฮฉ โ†’ โ„•} (hg : Measurable g) :
    Measurable (fun ฯ‰ โ†ฆ sumRewards A R' (f ฯ‰) (g ฯ‰) ฯ‰)
Type uses (1)
Body uses (1)
Used by (2)

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Proof
by
  change Measurable ((fun aฯ‰ โ†ฆ sumRewards A R' aฯ‰.1 (g aฯ‰.2) aฯ‰.2) โˆ˜ fun ฯ‰ โ†ฆ (f ฯ‰, ฯ‰))
  apply Measurable.comp _ (by fun_prop)
  refine measurable_from_prod_countable_right fun a โ†ฆ ?_
  change Measurable ((fun tฯ‰ โ†ฆ sumRewards A R' a tฯ‰.1 tฯ‰.2) โˆ˜ fun ฯ‰ โ†ฆ (g ฯ‰, ฯ‰))
  apply Measurable.comp _ (by fun_prop)
  exact measurable_from_prod_countable_right (fun t โ†ฆ measurable_sumRewards hA hR' a t)

Dependency graph

Type dependencies (1)

sumRewards๐Ÿ”—

DefinitionLearning.sumRewards
Details

Sum of rewards obtained when pulling action a up to time t (exclusive).

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

Code

def sumRewards (A : โ„• โ†’ ฮฉ โ†’ ๐“) (R' : โ„• โ†’ ฮฉ โ†’ โ„) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„ :=
  โˆ‘ s โˆˆ range t, if A s ฯ‰ = a then R' s ฯ‰ else 0
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