`Bandits.ArrayModel.prob_exists_pullCount_eq_and_sumRewards_mem_le`🔗

Lemma

No docstring.

theorem

Bandits.ArrayModel.prob_exists_pullCount_eq_and_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) (m : ℕ) {B : Set ℝ}
  (hB : MeasurableSet B) :
  (arrayMeasure ν)
      {ω |
        ∃ n,
          Learning.pullCount (action alg) a n ω = m ∧
            Learning.sumRewards (action alg) (reward alg) a n ω ∈ B} ≤
    (streamMeasure ν) {ω | ∑ i ∈ Finset.range m, ω i a ∈ B}
Bandits.ArrayModel.prob_exists_pullCount_eq_and_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) (m : ℕ) {B : Set ℝ}
  (hB : MeasurableSet B) :
  (arrayMeasure ν)
      {ω |
        ∃ n,
          Learning.pullCount (action alg)
                a n ω =
              m ∧
            Learning.sumRewards
                (action alg) (reward alg)
                a n ω ∈
              B} ≤
    (streamMeasure ν)
      {ω |
        ∑ i ∈ Finset.range m, ω i a ∈ B}

Code

lemma prob_exists_pullCount_eq_and_sumRewards_mem_le (a : 𝓐) (m : ℕ) {B : Set ℝ}
    (hB : MeasurableSet B) : 𝔓 {ω | ∃ n, pullCount A a n ω = m ∧ sumRewards A R a n ω ∈ B} ≤
      streamMeasure ν {ω | ∑ i ∈ range m, ω i a ∈ B}

Proof

calc
    _ ≤ 𝔓 {ω | ∑ i ∈ range m, ω.2 i a ∈ B} := by
        apply measure_mono
        intro ω ⟨n, hp, hn⟩
        rwa [sumRewards_eq alg a n ω, hp] at hn
    _ = streamMeasure ν {ω | ∑ i ∈ range m, ω i a ∈ B} :=
        (identDistrib_sum_range_snd a m).measure_mem_eq hB

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, 106 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_exists_pullCount_eq_and_sumRewards_mem_le🔗

`Bandits.ArrayModel.prob_exists_pullCount_eq_and_sumRewards_mem_le`🔗