LeanMachineLearning exposition

Learning.measurable_sumRewards๐Ÿ”—

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Minimal Lean file

measurable_sumRewards๐Ÿ”—

LemmaLearning.measurable_sumRewards
Details

No docstring.

๐Ÿ”—theorem
Learning.measurable_sumRewards.{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“} [MeasurableSingletonClass ๐“] {R' : โ„• โ†’ ฮฉ โ†’ โ„}
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hR' : โˆ€ (n : โ„•), Measurable (R' n)) (a : ๐“) (t : โ„•) :
  Measurable (sumRewards A R' a t)
Learning.measurable_sumRewards.{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“}
  [MeasurableSingletonClass ๐“]
  {R' : โ„• โ†’ ฮฉ โ†’ โ„}
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hR' : โˆ€ (n : โ„•), Measurable (R' n))
  (a : ๐“) (t : โ„•) :
  Measurable (sumRewards A R' a t)

Code

lemma measurable_sumRewards [MeasurableSingletonClass ๐“] {R' : โ„• โ†’ ฮฉ โ†’ โ„}
    (hA : โˆ€ n, Measurable (A n)) (hR' : โˆ€ n, Measurable (R' n)) (a : ๐“) (t : โ„•) :
    Measurable (sumRewards A R' a t)
Type uses (1)
Used by (6)

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Proof
by
  unfold sumRewards
  have h_meas s : Measurable (fun h : ฮฉ โ†ฆ if A s h = a then R' s h else 0) := by
    refine Measurable.ite ?_ (by fun_prop) (by fun_prop)
    exact (measurableSet_singleton _).preimage (by fun_prop)
  fun_prop

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
Used by (44)

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