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Learning.pullCount_add_one_eq_pullCount'๐Ÿ”—

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pullCount_add_one_eq_pullCount'๐Ÿ”—

LemmaLearning.pullCount_add_one_eq_pullCount'
Details

No docstring.

๐Ÿ”—theorem
Learning.pullCount_add_one_eq_pullCount'.{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“}
  {R' : โ„• โ†’ ฮฉ โ†’ R} {a : ๐“} {n : โ„•} {ฯ‰ : ฮฉ} :
  pullCount A a (n + 1) ฯ‰ =
    pullCount' n (fun i => (A (โ†‘i) ฯ‰, R' (โ†‘i) ฯ‰)) a
Learning.pullCount_add_one_eq_pullCount'.{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“} {R' : โ„• โ†’ ฮฉ โ†’ R} {a : ๐“}
  {n : โ„•} {ฯ‰ : ฮฉ} :
  pullCount A a (n + 1) ฯ‰ =
    pullCount' n
      (fun i => (A (โ†‘i) ฯ‰, R' (โ†‘i) ฯ‰)) a

Code

lemma pullCount_add_one_eq_pullCount' {n : โ„•} {ฯ‰ : ฮฉ} :
    pullCount A a (n + 1) ฯ‰ = pullCount' n (fun i โ†ฆ (A i ฯ‰, R' i ฯ‰)) a
Type uses (2)
Body uses (2)
Used by (5)

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Proof
by
  rw [pullCount_eq_sum, pullCount'_eq_sum]
  rw [Finset.sum_coe_sort (f := fun s โ†ฆ if A s ฯ‰ = a then 1 else 0) (Iic n)]
  congr with m
  simp only [mem_range, mem_Iic]
  grind

Dependency graph

Type dependencies (2)

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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pullCount'๐Ÿ”—

DefinitionLearning.pullCount'
Details

Number of pulls of arm a up to (and including) time n. This is the number of entries in h in which the arm is a.

๐Ÿ”—def
Learning.pullCount'.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  [DecidableEq ๐“] (n : โ„•) (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) (a : ๐“) : โ„•
Learning.pullCount'.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  [DecidableEq ๐“] (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) (a : ๐“) :
  โ„•

Code

noncomputable
def pullCount' (n : โ„•) (h : Iic n โ†’ ๐“ ร— R) (a : ๐“) := #{s | (h s).1 = a}
Used by (29)

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