LeanMachineLearning exposition

Learning.stepsUntil_eq_iff'🔗

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Minimal Lean file

stepsUntil_eq_iff'🔗

LemmaLearning.stepsUntil_eq_iff'
Details

No docstring.

🔗theorem
Learning.stepsUntil_eq_iff'.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} {a : 𝓐} {m : ℕ} {ω : Ω} (hm : m ≠ 0)
  (n : ℕ) :
  stepsUntil A a m ω = ↑n ↔ A n ω = a ∧ pullCount A a n ω = m - 1
Learning.stepsUntil_eq_iff'.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} {a : 𝓐}
  {m : ℕ} {ω : Ω} (hm : m ≠ 0) (n : ℕ) :
  stepsUntil A a m ω = ↑n ↔
    A n ω = a ∧ pullCount A a n ω = m - 1

Code

lemma stepsUntil_eq_iff' {ω : Ω} (hm : m ≠ 0) (n : ℕ) :
    stepsUntil A a m ω = n ↔ A n ω = a ∧ pullCount A a n ω = m - 1
Type uses (2)
Body uses (5)
Used by (1)

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Proof
by
  by_cases hn : n = 0
  · simp [hn, stepsUntil_eq_zero_iff, hm]
    grind
  rw [stepsUntil_eq_iff n]
  refine ⟨fun ⟨h1, h2⟩ ↦ ⟨?_, ?_⟩, fun ⟨h1, h2⟩ ↦ ⟨?_, fun k hk ↦ ?_⟩⟩
  · rw [pullCount_add_one] at h1
    specialize h2 (n - 1) (by lia)
    grind
  · rw [pullCount_add_one] at h1
    specialize h2 (n - 1) (by lia)
    grind
  · rw [pullCount_add_one, h1, h2]
    grind
  · rw [Nat.lt_iff_le_pred (by grind)]
    rw [← h2]
    refine monotone_pullCount a ω ?_
    grind

Dependency graph

Type dependencies (2)

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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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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