`Bandits.ArrayModel.hasCondDistrib_reward_zero`🔗

Lemma

No docstring.

theorem

Bandits.ArrayModel.hasCondDistrib_reward_zero.{u_1, u_2} {𝓐 : Type u_1}
  {R : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  [Nonempty 𝓐] [StandardBorelSpace 𝓐] [DecidableEq 𝓐] [Countable 𝓐]
  [StandardBorelSpace R] [Nonempty R] (alg : Learning.Algorithm 𝓐 R)
  (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν] :
  ProbabilityTheory.HasCondDistrib (reward alg 0) (action alg 0) ν
    (arrayMeasure ν)
Bandits.ArrayModel.hasCondDistrib_reward_zero.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐]
  [Countable 𝓐] [StandardBorelSpace R]
  [Nonempty R]
  (alg : Learning.Algorithm 𝓐 R)
  (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν] :
  ProbabilityTheory.HasCondDistrib
    (reward alg 0) (action alg 0) ν
    (arrayMeasure ν)

Code

lemma hasCondDistrib_reward_zero (alg : Algorithm 𝓐 R) (ν : Kernel 𝓐 R) [IsMarkovKernel ν] :
    HasCondDistrib (reward alg 0) (action alg 0) ν (arrayMeasure ν)

Proof

by
  refine hasCondDistrib_of_condDistrib_eq (by fun_prop) (by fun_prop) ?_
  refine (condDistrib_ae_eq_cond (by fun_prop) (by fun_prop)).trans ?_
  rw [Filter.EventuallyEq, ae_iff_of_countable]
  intro a ha
  simp only [reward_zero]
  calc ((arrayMeasure ν)[|action alg 0 ⁻¹' {a}]).map (fun ω ↦ ω.2 0 (action alg 0 ω))
  _ = ((arrayMeasure ν)[|action alg 0 ⁻¹' {a}]).map (fun ω ↦ ω.2 0 a) := by
    refine Measure.map_congr
      (ae_cond_of_forall_mem ((measurableSet_singleton _).preimage (by fun_prop)) ?_)
    intro x hx
    simp only [Set.mem_preimage, Set.mem_singleton_iff] at hx
    simp [hx]
  _ = ν a := by
    rw [cond_of_indepFun]
    · exact map_snd_apply_arrayMeasure 0 a
    · have : (fun ω ↦ ω.1 0) ⟂ᵢ[arrayMeasure ν] fun ω ↦ ω.2 0 a :=
        indepFun_fst_zero_snd_zero_action ν a
      rw [action_zero]
      exact this.comp (φ := initAlgFunction alg) (by fun_prop) measurable_id
    · fun_prop
    · fun_prop
    · simp
    · rwa [Measure.map_apply (by fun_prop) (by simp)] at ha

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 5th 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: 13 project declarations, 91 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.ArrayModel.hasCondDistrib_reward_zero🔗

`Bandits.ArrayModel.hasCondDistrib_reward_zero`🔗