`Bandits.condIndepFun_reward_stepsUntil_action'`🔗

Lemma

No docstring.

theorem

Bandits.condIndepFun_reward_stepsUntil_action'.{u_1, u_2} {𝓐 : Type u_1}
  {Ω : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {mΩ : MeasurableSpace Ω}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {alg : Learning.Algorithm 𝓐 ℝ} {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] [StandardBorelSpace 𝓐]
  [Nonempty 𝓐] [StandardBorelSpace Ω]
  (h : Learning.IsAlgEnvSeq A R alg (Learning.stationaryEnv ν) P)
  (a : 𝓐) (m n : ℕ) :
  ProbabilityTheory.CondIndepFun
    (MeasurableSpace.comap (A n) inferInstance) ⋯ (R n)
    (Set.indicator {ω | Learning.stepsUntil A a m ω = ↑n} fun x => 1) P
Bandits.condIndepFun_reward_stepsUntil_action'.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω} [DecidableEq 𝓐]
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  [StandardBorelSpace Ω]
  (h :
    Learning.IsAlgEnvSeq A R alg
      (Learning.stationaryEnv ν) P)
  (a : 𝓐) (m n : ℕ) :
  ProbabilityTheory.CondIndepFun
    (MeasurableSpace.comap (A n)
      inferInstance)
    ⋯ (R n)
    (Set.indicator
      {ω |
        Learning.stepsUntil A a m ω = ↑n}
      fun x => 1)
    P

Code

lemma condIndepFun_reward_stepsUntil_action' [StandardBorelSpace Ω]
    (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) (a : 𝓐) (m n : ℕ) :
    R n ⟂ᵢ[A n, h.measurable_action n; P] {ω | stepsUntil A a m ω = ↑n}.indicator (fun _ ↦ 1)

Proof

by
  -- the indicator of `stepsUntil ... = n` is a function of `hist (n-1)` and `action n`.
  -- It thus suffices to use the independence of `reward n` and `hist (n-1)` conditionally
  -- on `action n`.
  have hA := h.measurable_action
  have hR := h.measurable_feedback
  by_cases hn : n = 0
  · have h_indep : R 0 ⟂ᵢ[A 0, hA 0; P] A 0 :=
      condIndepFun_self_right (by fun_prop) (by fun_prop)
    simp only [hn]
    refine h_indep.of_measurable_right (hX := hA 0) ?_
    exact measurable_comap_indicator_stepsUntil_eq_zero a m
  · have h_indep : R n ⟂ᵢ[A n, hA n; P] fun ω ↦ (history A R (n - 1) ω, A n ω) :=
      IsAlgEnvSeq.condIndepFun_feedback_history_action_action' h n (by grind)
    refine h_indep.of_measurable_right (hX := hA n) ?_
    exact measurable_comap_indicator_stepsUntil_eq h a m 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: 8 project declarations, 72 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.

Bandits.condIndepFun_reward_stepsUntil_action'🔗

`Bandits.condIndepFun_reward_stepsUntil_action'`🔗