LeanMachineLearning exposition

Learning.stepsUntil_eq_congr🔗

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Minimal Lean file

stepsUntil_eq_congr🔗

LemmaLearning.stepsUntil_eq_congr
Details

No docstring.

🔗theorem
Learning.stepsUntil_eq_congr.{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' : Ω}
  (h_eq : ∀ i ≤ n, A i ω = A i ω') :
  stepsUntil A a m ω = ↑n ↔ stepsUntil A a m ω' = ↑n
Learning.stepsUntil_eq_congr.{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' : Ω}
  (h_eq : ∀ i ≤ n, A i ω = A i ω') :
  stepsUntil A a m ω = ↑n ↔
    stepsUntil A a m ω' = ↑n

Code

lemma stepsUntil_eq_congr {ω' : Ω} (h_eq : ∀ i ≤ n, A i ω = A i ω') :
    stepsUntil A a m ω = n ↔ stepsUntil A a m ω' = n
Type uses (1)
Body uses (3)

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Proof
by
  simp_rw [stepsUntil_eq_iff n]
  congr! 1
  · rw [pullCount_congr h_eq]
  · congr! 3 with k hk
    rw [pullCount_congr]
    grind

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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