`Learning.IsAlgEnvSeq.condExp_feedback_comp`🔗

Lemma

No docstring.

theorem

Learning.IsAlgEnvSeq.condExp_feedback_comp.{u_1, u_2, u_3, u_4}
  {Ω : Type u_1} {𝓐 : Type u_2} {𝓨 : Type u_3} {mΩ : MeasurableSpace Ω}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  [NormedAddCommGroup 𝓨] {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨}
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P]
  {alg : Algorithm 𝓐 𝓨} {env : Environment 𝓐 𝓨} {𝓩 : Type u_4}
  [NormedAddCommGroup 𝓩] [NormedSpace ℝ 𝓩] [CompleteSpace 𝓩]
  [StandardBorelSpace 𝓨] (h : IsAlgEnvSeq A Y alg env P) (n : ℕ)
  {g : 𝓨 → 𝓩} (hg : MeasureTheory.StronglyMeasurable g)
  (hint : MeasureTheory.Integrable (fun ω => g (Y (n + 1) ω)) P) :
  P[fun ω => g (Y (n + 1) ω) | ↑(filtrationAction h) (n + 1)] =ᵐ[P]
    fun ω =>
    ∫ (x : 𝓨),
      g x ∂(Environment.feedback env n) (history A Y n ω, A (n + 1) ω)
Learning.IsAlgEnvSeq.condExp_feedback_comp.{u_1,
    u_2, u_3, u_4}
  {Ω : Type u_1} {𝓐 : Type u_2}
  {𝓨 : Type u_3} {mΩ : MeasurableSpace Ω}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  [NormedAddCommGroup 𝓨] {A : ℕ → Ω → 𝓐}
  {Y : ℕ → Ω → 𝓨}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure P]
  {alg : Algorithm 𝓐 𝓨}
  {env : Environment 𝓐 𝓨} {𝓩 : Type u_4}
  [NormedAddCommGroup 𝓩] [NormedSpace ℝ 𝓩]
  [CompleteSpace 𝓩] [StandardBorelSpace 𝓨]
  (h : IsAlgEnvSeq A Y alg env P) (n : ℕ)
  {g : 𝓨 → 𝓩}
  (hg :
    MeasureTheory.StronglyMeasurable g)
  (hint :
    MeasureTheory.Integrable
      (fun ω => g (Y (n + 1) ω)) P) :
  P[fun ω => g (Y (n + 1) ω) |
      ↑(filtrationAction h) (n + 1)] =ᵐ[P]
    fun ω =>
    ∫ (x : 𝓨),
      g
        x ∂(Environment.feedback env n)
        (history A Y n ω, A (n + 1) ω)

Code

lemma IsAlgEnvSeq.condExp_feedback_comp {𝓩 : Type*} [NormedAddCommGroup 𝓩] [NormedSpace ℝ 𝓩]
    [CompleteSpace 𝓩] [StandardBorelSpace 𝓨]
    (h : IsAlgEnvSeq A Y alg env P) (n : ℕ)
    {g : 𝓨 → 𝓩} (hg : StronglyMeasurable g) (hint : Integrable (fun ω ↦ g (Y (n + 1) ω)) P) :
    P[fun ω ↦ g (Y (n + 1) ω) | h.filtrationAction (n + 1)] =ᵐ[P]
      fun ω ↦ (env.feedback n (history A Y n ω, A (n + 1) ω))[g]

Proof

by
  have hX : Measurable (fun ω ↦ (history A Y n ω, A (n + 1) ω)) :=
    (h.measurable_history n).prodMk (h.measurable_action (n + 1))
  rw [h.filtrationAction_eq_comap (n + 1) (by simp)]
  exact (h.hasCondDistrib_feedback n).condExp_comp_eq hX hg hint

Actions: Source · 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 · open it in the Lean web editor.

Dependency graph

Audit surface: 6 project declarations, 78 external constants

✓ 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.

Learning.IsAlgEnvSeq.condExp_feedback_comp🔗

`Learning.IsAlgEnvSeq.condExp_feedback_comp`🔗