`Learning.IsObliviousEnv.condIndepFun_feedback_history_action`🔗

Lemma

The feedback at time n + 1 is conditionally independent of the history up to time n given the action at time n + 1.

theorem

Learning.IsObliviousEnv.condIndepFun_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} [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  [StandardBorelSpace 𝓨] [Nonempty 𝓨] [StandardBorelSpace Ω]
  [IsObliviousEnv env] (h : IsAlgEnvSeq A Y alg env P) (n : ℕ) :
  ProbabilityTheory.CondIndepFun
    (MeasurableSpace.comap (A (n + 1)) inferInstance) ⋯ (Y (n + 1))
    (history A Y n) P
Learning.IsObliviousEnv.condIndepFun_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}
  [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  [StandardBorelSpace 𝓨] [Nonempty 𝓨]
  [StandardBorelSpace Ω]
  [IsObliviousEnv env]
  (h : IsAlgEnvSeq A Y alg env P)
  (n : ℕ) :
  ProbabilityTheory.CondIndepFun
    (MeasurableSpace.comap (A (n + 1))
      inferInstance)
    ⋯ (Y (n + 1)) (history A Y n) P

Code

lemma condIndepFun_feedback_history_action [StandardBorelSpace Ω]
        [IsObliviousEnv env] (h : IsAlgEnvSeq A Y alg env P) (n : ℕ) :
    Y (n + 1) ⟂ᵢ[A (n + 1), h.measurable_action _ ; P] history A Y n

Proof

by
  have hA := h.measurable_action
  have hY := h.measurable_feedback
  refine condIndepFun_of_exists_condDistrib_prod_ae_eq_prodMkLeft
    (η := feedbackCondAction env (n + 1))
    (by fun_prop) (by fun_prop) (by fun_prop) ?_
  refine HasCondDistrib.condDistrib_eq ?_
  rw [← feedback_eq_feedbackCondAction]
  exact h.hasCondDistrib_feedback n

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 4th recorded change.

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: 5 project declarations, 49 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.condIndepFun_feedback_history_action🔗

`Learning.IsObliviousEnv.condIndepFun_feedback_history_action`🔗