LeanMachineLearning exposition

Learning.pullCount'_mono๐Ÿ”—

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Minimal Lean file

pullCount'_mono๐Ÿ”—

LemmaLearning.pullCount'_mono
Details

No docstring.

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

Code

lemma pullCount'_mono {n m : โ„•} (hnm : n โ‰ค m) :
    pullCount' n (fun i โ†ฆ (A i ฯ‰, R' i ฯ‰)) a โ‰ค pullCount' m (fun i โ†ฆ (A i ฯ‰, R' i ฯ‰)) a
Type uses (1)
Body uses (3)

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Proof
by
  rw [โ† pullCount_add_one_eq_pullCount', โ† pullCount_add_one_eq_pullCount']
  exact pullCount_mono a (by lia) _

Dependency graph

Type dependencies (1)

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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