`Bandits.ArrayModel.prob_sumRewards_le_sumRewards_le`🔗

Lemma

No docstring.

theorem

Bandits.ArrayModel.prob_sumRewards_le_sumRewards_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] [Fintype 𝓐] (a : 𝓐)
  (n m₁ m₂ : ℕ) :
  (arrayMeasure ν)
      {ω |
        Learning.pullCount (action alg) (bestArm ν) n ω = m₁ ∧
          Learning.pullCount (action alg) a n ω = m₂ ∧
            Learning.sumRewards (action alg) (reward alg) (bestArm ν) n
                ω ≤
              Learning.sumRewards (action alg) (reward alg) a n ω} ≤
    (streamMeasure ν)
      {ω |
        ∑ i ∈ Finset.range m₁, ω i (bestArm ν) ≤
          ∑ i ∈ Finset.range m₂, ω i a}
Bandits.ArrayModel.prob_sumRewards_le_sumRewards_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]
  [Fintype 𝓐] (a : 𝓐) (n m₁ m₂ : ℕ) :
  (arrayMeasure ν)
      {ω |
        Learning.pullCount (action alg)
              (bestArm ν) n ω =
            m₁ ∧
          Learning.pullCount (action alg)
                a n ω =
              m₂ ∧
            Learning.sumRewards
                (action alg) (reward alg)
                (bestArm ν) n ω ≤
              Learning.sumRewards
                (action alg) (reward alg)
                a n ω} ≤
    (streamMeasure ν)
      {ω |
        ∑ i ∈ Finset.range m₁,
            ω i (bestArm ν) ≤
          ∑ i ∈ Finset.range m₂, ω i a}

Code

lemma prob_sumRewards_le_sumRewards_le [Fintype 𝓐] (a : 𝓐) (n m₁ m₂ : ℕ) :
    (𝔓) {ω | pullCount A (bestArm ν) n ω = m₁ ∧ pullCount A a n ω = m₂ ∧
        sumRewards A R (bestArm ν) n ω ≤ sumRewards A R a n ω} ≤
      streamMeasure ν
        {ω | ∑ i ∈ range m₁, ω i (bestArm ν) ≤ ∑ i ∈ range m₂, ω i a}

Proof

by
  simp_rw [sumRewards_eq]
  calc 𝔓 {ω | pullCount A (bestArm ν) n ω = m₁ ∧ pullCount A a n ω = m₂ ∧
      ∑ i ∈ range (pullCount A (bestArm ν) n ω), ω.2 i (bestArm ν) ≤
        ∑ i ∈ range (pullCount A a n ω), ω.2 i a}
  _ ≤ 𝔓 ((fun ω ↦ (∑ i ∈ range m₁, ω.2 i (bestArm ν), ∑ i ∈ range m₂, ω.2 i a)) ⁻¹'
        {p | p.1 ≤ p.2}) := by
      refine measure_mono fun ω hω ↦ ?_
      simp only [Set.preimage_ofPred_eq, Set.mem_ofPred_eq] at hω ⊢
      grind
  _ = streamMeasure ν
      {ω | ∑ i ∈ range m₁, ω i (bestArm ν) ≤ ∑ i ∈ range m₂, ω i a} := by
    rw [← Measure.snd_prod (μ := (Measure.infinitePi fun (_ : ℕ) ↦ (volume : Measure unitInterval)))
      (ν := streamMeasure ν), Measure.snd, Measure.map_apply (by fun_prop)]
    · rfl
    simp only [measurableSet_setOfPred]
    fun_prop

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: 16 project declarations, 126 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_sumRewards_le_sumRewards_le🔗

`Bandits.ArrayModel.prob_sumRewards_le_sumRewards_le`🔗