LeanMachineLearning exposition

Learning.action_eq_of_stepsUntil_eq_coe🔗

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Minimal Lean file

action_eq_of_stepsUntil_eq_coe🔗

LemmaLearning.action_eq_of_stepsUntil_eq_coe
Details

No docstring.

🔗theorem
Learning.action_eq_of_stepsUntil_eq_coe.{u_1, u_3} {𝓐 : Type u_1}
  {Ω : Type u_3} [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} {a : 𝓐} {m n : ℕ}
  {ω : Ω} (hm : m ≠ 0) (h : stepsUntil A a m ω = ↑n) : A n ω = a
Learning.action_eq_of_stepsUntil_eq_coe.{u_1,
    u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} {a : 𝓐}
  {m n : ℕ} {ω : Ω} (hm : m ≠ 0)
  (h : stepsUntil A a m ω = ↑n) :
  A n ω = a

Code

lemma action_eq_of_stepsUntil_eq_coe (hm : m ≠ 0) (h : stepsUntil A a m ω = n) :
    A n ω = a
Type uses (1)
Body uses (3)
Used by (2)

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Proof
by
  have : n = (stepsUntil A a m ω).toNat := by simp [h]
  rw [this]
  have h_exists : ∃ s, pullCount A a (s + 1) ω = m := exists_pullCount_eq (by simp [h])
  exact action_stepsUntil hm h_exists

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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