LeanMachineLearning exposition

Learning.sum_pullCount_mul๐Ÿ”—

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Minimal Lean file

sum_pullCount_mul๐Ÿ”—

LemmaLearning.sum_pullCount_mul
Details

No docstring.

๐Ÿ”—theorem
Learning.sum_pullCount_mul.{u_1, u_2, u_3} {๐“ : Type u_1} {R : Type u_2}
  {ฮฉ : Type u_3} [DecidableEq ๐“] {A : โ„• โ†’ ฮฉ โ†’ ๐“} [Fintype ๐“]
  [Semiring R] (ฯ‰ : ฮฉ) (f : ๐“ โ†’ R) (t : โ„•) :
  โˆ‘ a, โ†‘(pullCount A a t ฯ‰) * f a = โˆ‘ s โˆˆ Finset.range t, f (A s ฯ‰)
Learning.sum_pullCount_mul.{u_1, u_2, u_3}
  {๐“ : Type u_1} {R : Type u_2}
  {ฮฉ : Type u_3} [DecidableEq ๐“]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} [Fintype ๐“] [Semiring R]
  (ฯ‰ : ฮฉ) (f : ๐“ โ†’ R) (t : โ„•) :
  โˆ‘ a, โ†‘(pullCount A a t ฯ‰) * f a =
    โˆ‘ s โˆˆ Finset.range t, f (A s ฯ‰)

Code

lemma sum_pullCount_mul [Fintype ๐“] [Semiring R] (ฯ‰ : ฮฉ) (f : ๐“ โ†’ R) (t : โ„•) :
    โˆ‘ a, pullCount A a t ฯ‰ * f a = โˆ‘ s โˆˆ range t, f (A s ฯ‰)
Type uses (1)
Used by (2)

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Proof
by
  unfold pullCount
  classical
  simp_rw [card_eq_sum_ones]
  push_cast
  simp_rw [sum_mul, one_mul]
  exact sum_fiberwise' (range t) (A ยท ฯ‰) f

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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