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} (n : ℕ) :
  stepsUntil A a m ω = ↑n ↔
    pullCount A a (n + 1) ω = m ∧ ∀ k < n, pullCount A a (k + 1) ω < m
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} (n : ℕ) :
  stepsUntil A a m ω = ↑n ↔
    pullCount A a (n + 1) ω = m ∧
      ∀ k < n, pullCount A a (k + 1) ω < m

Code

lemma stepsUntil_eq_iff {ω : Ω} (n : ℕ) :
    stepsUntil A a m ω = n ↔
      pullCount A a (n + 1) ω = m ∧ (∀ k < n, pullCount A a (k + 1) ω < m)
Type uses (2)
Body uses (4)
Used by (2)

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Proof
by
  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
  · have h_exists : ∃ s, pullCount A a (s + 1) ω = m := exists_pullCount_eq (by simp [h])
    refine ⟨pullCount_add_one_eq_of_stepsUntil_eq_coe h, fun k hk ↦ ?_⟩
    exact pullCount_lt_of_le_stepsUntil a ω h_exists (by rw [h]; exact mod_cast hk)
  · classical
    rw [stepsUntil_eq_dite a m ω, dif_pos ⟨n, h.1⟩]
    simp only [Nat.cast_inj]
    rw [Nat.find_eq_iff]
    exact ⟨h.1, fun k hk ↦ (h.2 k hk).ne⟩

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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