LeanMachineLearning exposition

Learning.pullCount_stepsUntil๐Ÿ”—

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Minimal Lean file

pullCount_stepsUntil๐Ÿ”—

LemmaLearning.pullCount_stepsUntil
Details

No docstring.

๐Ÿ”—theorem
Learning.pullCount_stepsUntil.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] {A : โ„• โ†’ ฮฉ โ†’ ๐“} {a : ๐“} {m : โ„•} {ฯ‰ : ฮฉ} (hm : m โ‰  0)
  (h_exists : โˆƒ s, pullCount A a (s + 1) ฯ‰ = m) :
  pullCount A a (ENat.toNat (stepsUntil A a m ฯ‰)) ฯ‰ = m - 1
Learning.pullCount_stepsUntil.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] {A : โ„• โ†’ ฮฉ โ†’ ๐“} {a : ๐“}
  {m : โ„•} {ฯ‰ : ฮฉ} (hm : m โ‰  0)
  (h_exists :
    โˆƒ s, pullCount A a (s + 1) ฯ‰ = m) :
  pullCount A a
      (ENat.toNat (stepsUntil A a m ฯ‰))
      ฯ‰ =
    m - 1

Code

lemma pullCount_stepsUntil (hm : m โ‰  0) (h_exists : โˆƒ s, pullCount A a (s + 1) ฯ‰ = m) :
    pullCount A a (stepsUntil A a m ฯ‰).toNat ฯ‰ = m - 1
Type uses (2)
Body uses (4)
Used by (1)

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Proof
by
  have h_action := action_eq_of_stepsUntil_eq_coe (A := A) (n := (stepsUntil A a m ฯ‰).toNat)
    (a := a) (ฯ‰ := ฯ‰) hm ?_
  swap; ยท symm; simpa [stepsUntil_eq_top_iff]
  have h_add_one := pullCount_stepsUntil_add_one h_exists
  nth_rw 1 [โ† h_action] at h_add_one
  rw [ENat.toNat_add ?_ (by simp), ENat.toNat_one, pullCount_action_eq_pullCount_add_one]
    at h_add_one
  swap; ยท simpa [stepsUntil_eq_top_iff]
  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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stepsUntil๐Ÿ”—

DefinitionLearning.stepsUntil
Details

Number of steps until action a was pulled exactly m times.

๐Ÿ”—def
Learning.stepsUntil.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (m : โ„•) (ฯ‰ : ฮฉ) : โ„•โˆž
Learning.stepsUntil.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“)
  (m : โ„•) (ฯ‰ : ฮฉ) : โ„•โˆž

Code

noncomputable
def stepsUntil (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (m : โ„•) (ฯ‰ : ฮฉ) : โ„•โˆž :=
  sInf ((โ†‘) '' {s | pullCount A a (s + 1) ฯ‰ = m})
Body uses (1)
Used by (46)

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