LeanMachineLearning exposition

Learning.stepsUntil_zero_of_ne🔗

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Minimal Lean file

stepsUntil_zero_of_ne🔗

LemmaLearning.stepsUntil_zero_of_ne
Details

No docstring.

🔗theorem
Learning.stepsUntil_zero_of_ne.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} {a : 𝓐} {ω : Ω} (hka : A 0 ω ≠ a) :
  stepsUntil A a 0 ω = 0
Learning.stepsUntil_zero_of_ne.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} {a : 𝓐}
  {ω : Ω} (hka : A 0 ω ≠ a) :
  stepsUntil A a 0 ω = 0

Code

lemma stepsUntil_zero_of_ne (hka : A 0 ω ≠ a) : stepsUntil A a 0 ω = 0
Type uses (1)
Body uses (3)
Used by (3)

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Proof
by
  unfold stepsUntil
  simp_rw [← bot_eq_zero, sInf_eq_bot, bot_eq_zero]
  intro n hn
  refine ⟨0, ?_, hn⟩
  simp only [Set.mem_image, Set.mem_setOf_eq, Nat.cast_eq_zero, exists_eq_right, zero_add]
  rw [← zero_add 1, pullCount_eq_pullCount_of_action_ne hka]
  simp

Dependency graph

Type dependencies (1)

stepsUntil🔗

DefinitionLearning.stepsUntil
Details

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

🔗def
Learning.stepsUntil.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐) (m : ℕ) (ω : Ω) : ℕ∞
Learning.stepsUntil.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐)
  (m : ℕ) (ω : Ω) : ℕ∞

Code

noncomputable
def stepsUntil (A : ℕ → Ω → 𝓐) (a : 𝓐) (m : ℕ) (ω : Ω) : ℕ∞ :=
  sInf ((↑) '' {s | pullCount A a (s + 1) ω = m})
Body uses (1)
Used by (46)

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All dependencies, transitively (1)

pullCount🔗

DefinitionLearning.pullCount
Details

Number of times action a was chosen up to time t (excluding t).

🔗def
Learning.pullCount.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐) (t : ℕ) (ω : Ω) : ℕ
Learning.pullCount.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐)
  (t : ℕ) (ω : Ω) : ℕ

Code

noncomputable
def pullCount (A : ℕ → Ω → 𝓐) (a : 𝓐) (t : ℕ) (ω : Ω) : ℕ :=
  #(filter (fun s ↦ A s ω = a) (range t))
Used by (146)

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