`Bandits.ArrayModel.prob_pullCount_prod_sumRewards_mem_le`🔗

Lemma

No docstring.

theorem

Bandits.ArrayModel.prob_pullCount_prod_sumRewards_mem_le.{u_1}
  {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐} [DecidableEq 𝓐] [Countable 𝓐]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐] {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] (a : 𝓐) (n : ℕ) {s : Set (ℕ × ℝ)}
  [DecidablePred fun x => x ∈ Prod.fst '' s] (hs : MeasurableSet s) :
  (arrayMeasure ν)
      {ω |
        (Learning.pullCount (action alg) a n ω,
            Learning.sumRewards (action alg) (reward alg) 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.ArrayModel.prob_pullCount_prod_sumRewards_mem_le.{u_1}
  {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  [DecidableEq 𝓐] [Countable 𝓐]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  (a : 𝓐) (n : ℕ) {s : Set (ℕ × ℝ)}
  [DecidablePred fun x =>
      x ∈ Prod.fst '' s]
  (hs : MeasurableSet s) :
  (arrayMeasure ν)
      {ω |
        (Learning.pullCount (action alg) a
              n ω,
            Learning.sumRewards
              (action alg) (reward alg) 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 (a : 𝓐) (n : ℕ)
    {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
  simp_rw [sumRewards_eq]
  calc 𝔓 ((fun ω ↦ (pullCount A a n ω, ∑ i ∈ range (pullCount A a n ω), ω.2 i a)) ⁻¹' s)
  _ ≤ 𝔓 {ω | ∃ k ≤ n, (k, ∑ i ∈ range k, ω.2 i a) ∈ s} := by
    refine measure_mono fun ω hω ↦ ?_
    simp only [Set.mem_ofPred_eq] at hω ⊢
    exact ⟨pullCount A a n ω, pullCount_le _ _ _, hω⟩
  _ = 𝔓 (⋃ k ∈ (range (n + 1)).filter (· ∈ Prod.fst '' s),
      {ω | (k, ∑ i ∈ range k, ω.2 i a) ∈ s}) := by congr 1; ext; simp; grind
  _ ≤ ∑ k ∈ (range (n + 1)).filter (· ∈ Prod.fst '' s),
      𝔓 {ω | ∑ i ∈ range k, ω.2 i a ∈ Prod.mk k ⁻¹' s} := measure_biUnion_finset_le _ _
  _ = ∑ k ∈ (range (n + 1)).filter (· ∈ Prod.fst '' s),
      (streamMeasure ν) {ω | ∑ i ∈ range k, ω i a ∈ Prod.mk k ⁻¹' s} := by
    congr with k
    have : (𝔓).map (fun ω ↦ ∑ i ∈ range k, ω.2 i a) =
        (streamMeasure ν).map (fun ω ↦ ∑ i ∈ range k, ω i a) :=
      (identDistrib_sum_range_snd a k).map_eq
    rw [Measure.ext_iff] at this
    specialize this (Prod.mk k ⁻¹' s) (hs.preimage (by fun_prop))
    rwa [Measure.map_apply (by fun_prop) (hs.preimage (by fun_prop)),
      Measure.map_apply (by fun_prop) (hs.preimage (by fun_prop))] at this

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-14-gf86702d (2026-08-25), the 6th 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: 15 project declarations, 111 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.prob_pullCount_prod_sumRewards_mem_le🔗

`Bandits.ArrayModel.prob_pullCount_prod_sumRewards_mem_le`🔗