`Learning.IsObliviousEnv.hasCondDistrib_feedback_history_action`🔗

Lemma

No docstring.

theorem

Learning.IsObliviousEnv.hasCondDistrib_feedback_history_action.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {Ω : Type u_3} {mΩ : MeasurableSpace Ω}
  {alg : Algorithm 𝓐 𝓨} {env : Environment 𝓐 𝓨}
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P]
  {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨} [IsObliviousEnv env]
  (h : IsAlgEnvSeq A Y alg env P) (n : ℕ) :
  ProbabilityTheory.HasCondDistrib (Y (n + 1))
    (fun ω => (history A Y n ω, A (n + 1) ω))
    (ProbabilityTheory.Kernel.prodMkLeft (↥(Finset.Iic n) → 𝓐 × 𝓨)
      (feedbackCondAction env (n + 1)))
    P
Learning.IsObliviousEnv.hasCondDistrib_feedback_history_action.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {Ω : Type u_3}
  {mΩ : MeasurableSpace Ω}
  {alg : Algorithm 𝓐 𝓨}
  {env : Environment 𝓐 𝓨}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure P]
  {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨}
  [IsObliviousEnv env]
  (h : IsAlgEnvSeq A Y alg env P)
  (n : ℕ) :
  ProbabilityTheory.HasCondDistrib
    (Y (n + 1))
    (fun ω =>
      (history A Y n ω, A (n + 1) ω))
    (ProbabilityTheory.Kernel.prodMkLeft
      (↥(Finset.Iic n) → 𝓐 × 𝓨)
      (feedbackCondAction env (n + 1)))
    P

Code

lemma hasCondDistrib_feedback_history_action [IsObliviousEnv env]
    (h : IsAlgEnvSeq A Y alg env P) (n : ℕ) :
    HasCondDistrib (Y (n + 1)) (fun ω ↦ (history A Y n ω, A (n + 1) ω))
      ((feedbackCondAction env (n + 1)).prodMkLeft _) P

Proof

by
  have hA := h.measurable_action
  have hR' := h.measurable_feedback
  refine ⟨by fun_prop, ?_⟩
  have h_eq := (h.hasCondDistrib_feedback n).map_eq
  simpa only [feedback_eq_feedbackCondAction] using h_eq

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, 44 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.IsObliviousEnv.hasCondDistrib_feedback_history_action🔗

`Learning.IsObliviousEnv.hasCondDistrib_feedback_history_action`🔗