`Bandits.prob_pullCount_prod_sumRewards_mem_le`🔗

Lemma

No docstring.

theorem

Bandits.prob_pullCount_prod_sumRewards_mem_le.{u_1, u_2} {𝓐 : Type u_1}
  {Ω : Type u_2} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  {n : ℕ} {a : 𝓐} [Nonempty 𝓐] [Countable 𝓐]
  [MeasurableSingletonClass 𝓐]
  (h : Learning.IsAlgEnvSeq A R alg (Learning.stationaryEnv ν) P)
  {s : Set (ℕ × ℝ)} [DecidablePred fun x => x ∈ Prod.fst '' s]
  (hs : MeasurableSet s) :
  P
      {ω |
        (Learning.pullCount A a n ω, Learning.sumRewards A R a n ω) ∈
          s} ≤
    ∑ k ∈ Finset.range (n + 1) with k ∈ Prod.fst '' s,
      (streamMeasure ν)
        {ω | ∑ i ∈ Finset.range k, ω i a ∈ Prod.mk k ⁻¹' s}
Bandits.prob_pullCount_prod_sumRewards_mem_le.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ} {n : ℕ}
  {a : 𝓐} [Nonempty 𝓐] [Countable 𝓐]
  [MeasurableSingletonClass 𝓐]
  (h :
    Learning.IsAlgEnvSeq A R alg
      (Learning.stationaryEnv ν) P)
  {s : Set (ℕ × ℝ)}
  [DecidablePred fun x =>
      x ∈ Prod.fst '' s]
  (hs : MeasurableSet s) :
  P
      {ω |
        (Learning.pullCount A a n ω,
            Learning.sumRewards A R a n
              ω) ∈
          s} ≤
    ∑ k ∈ Finset.range (n + 1) with
      k ∈ Prod.fst '' s,
      (streamMeasure ν)
        {ω |
          ∑ i ∈ Finset.range k, ω i a ∈
            Prod.mk k ⁻¹' s}

Code

lemma prob_pullCount_prod_sumRewards_mem_le [Countable 𝓐] [MeasurableSingletonClass 𝓐]
    (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P)
    {s : Set (ℕ × ℝ)} [DecidablePred (· ∈ Prod.fst '' s)] (hs : MeasurableSet s) :
    P {ω | (pullCount A a n ω, sumRewards A R a n ω) ∈ s} ≤
      ∑ k ∈ (range (n + 1)).filter (· ∈ Prod.fst '' s),
        streamMeasure ν {ω | ∑ i ∈ range k, ω i a ∈ Prod.mk k ⁻¹' s}

Proof

by
  have hA := h.measurable_action
  have hR := h.measurable_feedback
  calc P {ω | (pullCount A a n ω, sumRewards A R a n ω) ∈ s}
  _ = (P.map (fun ω ↦ (pullCount A a n ω, sumRewards A R a n ω))) s := by
      rw [Measure.map_apply (by fun_prop) hs]; rfl
  _ = ((ArrayModel.arrayMeasure ν).map
      (fun ω ↦ (pullCount (ArrayModel.action alg) a n ω,
        sumRewards (ArrayModel.action alg) (ArrayModel.reward alg) a n ω))) s := by
    rw [h.law_pullCount_sumRewards_unique (ArrayModel.isAlgEnvSeq_arrayMeasure alg ν)]
  _ = (ArrayModel.arrayMeasure ν) {ω | (pullCount (ArrayModel.action alg) a n ω,
      sumRewards (ArrayModel.action alg) (ArrayModel.reward alg) a n ω) ∈ s} := by
    rw [Measure.map_apply (by fun_prop) hs]; rfl
  _ ≤ ∑ k ∈ (range (n + 1)).filter (· ∈ Prod.fst '' s),
      streamMeasure ν {ω | ∑ i ∈ range k, ω i a ∈ Prod.mk k ⁻¹' s} :=
    ArrayModel.prob_pullCount_prod_sumRewards_mem_le a n hs

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Meaning last changed in v4.34.0-rc2-14-gf86702d (2026-08-25), 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: 9 project declarations, 81 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.prob_pullCount_prod_sumRewards_mem_le🔗

`Bandits.prob_pullCount_prod_sumRewards_mem_le`🔗