LeanMachineLearning

Bandits.ArrayModel.measurable_histπŸ”—

Lemma

No docstring.

πŸ”—theorem
Bandits.ArrayModel.measurable_hist.{u_1, u_2} {𝓐 : Type u_1}
  {R : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  [Nonempty 𝓐] [StandardBorelSpace 𝓐] [DecidableEq 𝓐] [Countable 𝓐]
  (alg : Learning.Algorithm 𝓐 R) (n : β„•) :
  Measurable fun Ο‰ => hist alg Ο‰ n
Bandits.ArrayModel.measurable_hist.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐]
  [Countable 𝓐]
  (alg : Learning.Algorithm 𝓐 R) (n : β„•) :
  Measurable fun Ο‰ => hist alg Ο‰ n

Code

lemma measurable_hist [DecidableEq 𝓐] [Countable 𝓐] (alg : Algorithm 𝓐 R) (n : β„•) :
    Measurable (fun Ο‰ ↦ hist alg Ο‰ n)
Proof
by
  induction n with
  | zero =>
    simp_rw [hist_zero, measurable_pi_iff]
    refine fun _ ↦ Measurable.prodMk (by fun_prop) ?_
    change Measurable ((fun x : 𝓐 Γ— ((β„• β†’ I) Γ— (β„• β†’ 𝓐 β†’ R)) ↦ x.2.2 0 x.1) ∘
        (fun x : (β„• β†’ I) Γ— (β„• β†’ 𝓐 β†’ R) ↦ (initAlgFunction alg (x.1 0), x)))
    have : Measurable (fun x : 𝓐 Γ— ((β„• β†’ I) Γ— (β„• β†’ 𝓐 β†’ R)) ↦ x.2.2 0 x.1) :=
      measurable_from_prod_countable_right fun p ↦ by simp only; fun_prop
    exact Measurable.comp (by fun_prop) (Measurable.prodMk (by fun_prop) (by fun_prop))
  | succ n hn =>
    refine measurable_pi_iff.mpr fun i ↦ ?_
    by_cases hin : i ≀ n
    Β· simp only [hist, hin, ↓reduceDIte]
      rw [measurable_pi_iff] at hn
      exact hn ⟨i.1, by simp [hin]⟩
    Β· simp only [hist, hin, ↓reduceDIte]
      refine Measurable.prodMk (by fun_prop) ?_
      change Measurable ((fun (x : (β„• β†’ 𝓐 β†’ R) Γ— β„• Γ— 𝓐) ↦ x.1 x.2.1 x.2.2) ∘
        (fun x ↦ (x.2, pullCount' n (hist alg x n) (algFunction alg n (hist alg x n) (x.1 (n + 1))),
          (algFunction alg n (hist alg x n) (x.1 (n + 1))))))
      have h1 : Measurable (fun (x : (β„• β†’ 𝓐 β†’ R) Γ— β„• Γ— 𝓐) ↦ x.1 x.2.1 x.2.2) :=
        measurable_from_prod_countable_left fun p : β„• Γ— 𝓐 ↦ (by simp only; fun_prop)
      refine Measurable.comp (by fun_prop) (Measurable.prodMk (by fun_prop) ?_)
      refine Measurable.prodMk ?_ (by fun_prop)
      exact measurable_pullCount'_action_add_one n hn

Actions: Source Β· Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 5th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file Β· open it in the Lean web editor.

Dependency graph

Audit surface: 9 project declarations, 88 external constants

βœ“ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.