LeanMachineLearning

Learning.IsAlgEnvSeq.integrable_means_actionπŸ”—

Lemma

No docstring.

πŸ”—theorem
Learning.IsAlgEnvSeq.integrable_means_action.{u_1, u_2, u_3}
  {Ξ© : Type u_1} {𝓐 : Type u_2} {𝓨 : Type u_3} {mΞ© : MeasurableSpace Ξ©}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  [NormedAddCommGroup 𝓨] [NormedSpace ℝ 𝓨] {A : β„• β†’ Ξ© β†’ 𝓐}
  {Y : β„• β†’ Ξ© β†’ 𝓨} {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsFiniteMeasure P] {alg : Algorithm 𝓐 𝓨}
  {env : Environment 𝓐 𝓨} [SecondCountableTopology 𝓨]
  [OpensMeasurableSpace 𝓨] (h : IsAlgEnvSeq A Y alg env P) {n : β„•}
  (hint : MeasureTheory.Integrable (Y n) P) :
  MeasureTheory.Integrable
    (fun Ο‰ => Environment.means env A Y (A n Ο‰) n Ο‰) P
Learning.IsAlgEnvSeq.integrable_means_action.{u_1,
    u_2, u_3}
  {Ξ© : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} {mΞ© : MeasurableSpace Ξ©}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  [NormedAddCommGroup 𝓨] [NormedSpace ℝ 𝓨]
  {A : β„• β†’ Ξ© β†’ 𝓐} {Y : β„• β†’ Ξ© β†’ 𝓨}
  {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsFiniteMeasure P]
  {alg : Algorithm 𝓐 𝓨}
  {env : Environment 𝓐 𝓨}
  [SecondCountableTopology 𝓨]
  [OpensMeasurableSpace 𝓨]
  (h : IsAlgEnvSeq A Y alg env P) {n : β„•}
  (hint :
    MeasureTheory.Integrable (Y n) P) :
  MeasureTheory.Integrable
    (fun Ο‰ =>
      Environment.means env A Y (A n Ο‰) n
        Ο‰)
    P

Code

lemma IsAlgEnvSeq.integrable_means_action [SecondCountableTopology 𝓨] [OpensMeasurableSpace 𝓨]
    (h : IsAlgEnvSeq A Y alg env P) {n : β„•} (hint : Integrable (Y n) P) :
    Integrable (fun Ο‰ ↦ env.means A Y (A n Ο‰) n Ο‰) P
Proof
by
  have hA := h.measurable_action
  have h_hist := h.measurable_history
  have hint' : Integrable id (P.map (Y n)) := by
    rwa [integrable_map_measure (by fun_prop) (h.measurable_feedback _).aemeasurable]
  unfold Environment.means Environment.measure
  cases n with
  | zero =>
    simp only [↓reduceIte, id_eq]
    rw [h.hasLaw_feedback_zero_comp.map_eq, Measure.integrable_comp_iff (by fun_prop)] at hint'
    have hint'' := hint'.2.comp_aemeasurable (by fun_prop)
    simp only [id_eq] at hint''
    refine Integrable.mono' hint'' ?_ ?_
    Β· refine StronglyMeasurable.aestronglyMeasurable ?_
      have h_eq Ο‰ : env.Ξ½0 (A 0 Ο‰) =
          (env.Ξ½0 βˆ˜β‚– Kernel.deterministic (A 0) (by fun_prop)) Ο‰ := by
        simp [Kernel.comp_deterministic_eq_comap]
      simp_rw [h_eq]
      exact StronglyMeasurable.integral_kernel (by fun_prop)
    Β· simp only [Function.comp_apply]
      filter_upwards with Ο‰ using norm_integral_le_integral_norm _
  | succ n =>
    simp only [Nat.add_eq_zero_iff, one_ne_zero, and_false, ↓reduceIte, Nat.add_one_sub_one, id_eq]
    rw [(h.hasLaw_feedback_comp n).map_eq, Measure.integrable_comp_iff (by fun_prop)] at hint'
    have hint'' := hint'.2.comp_aemeasurable (by fun_prop)
    simp only [id_eq] at hint''
    refine Integrable.mono' hint'' ?_ ?_
    Β· refine StronglyMeasurable.aestronglyMeasurable ?_
      have h_eq Ο‰ : (env.feedback n) (history A Y n Ο‰, A (n + 1) Ο‰) =
          (env.feedback n βˆ˜β‚–
            Kernel.deterministic (fun Ο‰ ↦ (history A Y n Ο‰, A (n + 1) Ο‰)) (by fun_prop)) Ο‰ := by
        simp [Kernel.comp_deterministic_eq_comap]
      simp_rw [Kernel.sectR_apply, h_eq]
      exact StronglyMeasurable.integral_kernel (by fun_prop)
    Β· simp only [Function.comp_apply]
      filter_upwards with Ο‰ using norm_integral_le_integral_norm _

Actions: Source Β· Open Issue

New in v4.34.0-rc2-14-gf86702d (2026-08-25), and its meaning has not changed since.

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: 6 project declarations, 63 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.