LeanMachineLearning exposition

Learning.sumRewards_add_one๐Ÿ”—

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Minimal Lean file

sumRewards_add_one๐Ÿ”—

LemmaLearning.sumRewards_add_one
Details

No docstring.

๐Ÿ”—theorem
Learning.sumRewards_add_one.{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}
  {R' : โ„• โ†’ ฮฉ โ†’ โ„} :
  sumRewards A R' a (t + 1) ฯ‰ =
    sumRewards A R' a t ฯ‰ + if A t ฯ‰ = a then R' t ฯ‰ else 0
Learning.sumRewards_add_one.{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} {R' : โ„• โ†’ ฮฉ โ†’ โ„} :
  sumRewards A R' a (t + 1) ฯ‰ =
    sumRewards A R' a t ฯ‰ +
      if A t ฯ‰ = a then R' t ฯ‰ else 0

Code

lemma sumRewards_add_one {R' : โ„• โ†’ ฮฉ โ†’ โ„} :
    sumRewards A R' a (t + 1) ฯ‰ = sumRewards A R' a t ฯ‰ + if A t ฯ‰ = a then R' t ฯ‰ else 0
Type uses (1)
Used by (3)

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Proof
by
  unfold sumRewards
  rw [sum_range_succ]

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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