LeanMachineLearning exposition

Learning.stepsUntil_mono🔗

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Minimal Lean file

stepsUntil_mono🔗

LemmaLearning.stepsUntil_mono
Details

No docstring.

🔗theorem
Learning.stepsUntil_mono.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} (a : 𝓐) (ω : Ω) {n m : ℕ} (hn : n ≠ 0)
  (hnm : n ≤ m) : stepsUntil A a n ω ≤ stepsUntil A a m ω
Learning.stepsUntil_mono.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} (a : 𝓐)
  (ω : Ω) {n m : ℕ} (hn : n ≠ 0)
  (hnm : n ≤ m) :
  stepsUntil A a n ω ≤ stepsUntil A a m ω

Code

lemma stepsUntil_mono (a : 𝓐) (ω : Ω) {n m : ℕ} (hn : n ≠ 0) (hnm : n ≤ m) :
    stepsUntil A a n ω ≤ stepsUntil A a m ω
Type uses (1)
Body uses (2)

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Proof
by
  rw [stepsUntil_eq_leastGE a hn, stepsUntil_eq_leastGE a (by lia)]
  simp_rw [leastGE]
  exact hittingAfter_anti (fun n ω ↦ (pullCount A a (n + 1) ω)) 0 (fun x ↦ by grind) ω

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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