LeanMachineLearning exposition

Learning.pullCount_stepsUntil_add_one๐Ÿ”—

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Minimal Lean file

pullCount_stepsUntil_add_one๐Ÿ”—

LemmaLearning.pullCount_stepsUntil_add_one
Details

No docstring.

๐Ÿ”—theorem
Learning.pullCount_stepsUntil_add_one.{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}
  (h_exists : โˆƒ s, pullCount A a (s + 1) ฯ‰ = m) :
  pullCount A a (ENat.toNat (stepsUntil A a m ฯ‰ + 1)) ฯ‰ = m
Learning.pullCount_stepsUntil_add_one.{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}
  (h_exists :
    โˆƒ s, pullCount A a (s + 1) ฯ‰ = m) :
  pullCount A a
      (ENat.toNat
        (stepsUntil A a m ฯ‰ + 1))
      ฯ‰ =
    m

Code

lemma pullCount_stepsUntil_add_one (h_exists : โˆƒ s, pullCount A a (s + 1) ฯ‰ = m) :
    pullCount A a (stepsUntil A a m ฯ‰ + 1).toNat ฯ‰ = m
Type uses (2)
Body uses (1)
Used by (3)

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Proof
by
  classical
  have h_eq := stepsUntil_eq_dite (A := A) a m ฯ‰
  simp only [h_exists, โ†“reduceDIte] at h_eq
  have h' := Nat.find_spec h_exists
  rw [h_eq]
  rw [ENat.toNat_add (by simp) (by simp)]
  simp only [ENat.toNat_coe, ENat.toNat_one]
  exact h'

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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