Bandits.UCB.some_sum_eq_zero.{u_1} {K : ℕ} {hK : 0 < K} {c : ℝ}
  {ν : ProbabilityTheory.Kernel (Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] {Ω : Type u_1}
  {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {A : ℕ → Ω → Fin K}
  {R : ℕ → Ω → ℝ} {σ2 : NNReal} [Nonempty (Fin K)]
  (h :
    Learning.IsAlgEnvSeq A R (ucbAlgorithm hK (c * ↑σ2))
      (Learning.stationaryEnv ν) P)
  (hc : 0 ≤ c) (a : Fin K) (h_gap : 0 < gap ν a) (n C : ℕ) (hC : C ≠ 0)
  (hC' : 8 * c * ↑σ2 * Real.log (↑n + 1) / gap ν a ^ 2 ≤ ↑C) :
  ∀ᵐ (ω : Ω) ∂P,
    ∑ s ∈ Finset.range n,
        Set.indicator
          {s |
            A s ω = a ∧
              C < Learning.pullCount A a s ω ∧
                ∫ (x : ℝ), id x ∂ν (bestArm ν) ≤
                    Learning.empMean A R (bestArm ν) s ω +
                      ucbWidth A (c * ↑σ2) (bestArm ν) s ω ∧
                  Learning.empMean A R (A s ω) s ω -
                      ucbWidth A (c * ↑σ2) (A s ω) s ω ≤
                    ∫ (x : ℝ), id x ∂ν (A s ω)}
          1 s =
      0
Bandits.UCB.some_sum_eq_zero.{u_1} {K : ℕ}
  {hK : 0 < K} {c : ℝ}
  {ν : ProbabilityTheory.Kernel (Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  {Ω : Type u_1} {mΩ : MeasurableSpace Ω}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → Fin K} {R : ℕ → Ω → ℝ}
  {σ2 : NNReal} [Nonempty (Fin K)]
  (h :
    Learning.IsAlgEnvSeq A R
      (ucbAlgorithm hK (c * ↑σ2))
      (Learning.stationaryEnv ν) P)
  (hc : 0 ≤ c) (a : Fin K)
  (h_gap : 0 < gap ν a) (n C : ℕ)
  (hC : C ≠ 0)
  (hC' :
    8 * c * ↑σ2 * Real.log (↑n + 1) /
        gap ν a ^ 2 ≤
      ↑C) :
  ∀ᵐ (ω : Ω) ∂P,
    ∑ s ∈ Finset.range n,
        Set.indicator
          {s |
            A s ω = a ∧
              C <
                  Learning.pullCount A a s
                    ω ∧
                ∫ (x : ℝ),
                      id
                        x ∂ν (bestArm ν) ≤
                    Learning.empMean A R
                        (bestArm ν) s ω +
                      ucbWidth A (c * ↑σ2)
                        (bestArm ν) s ω ∧
                  Learning.empMean A R
                        (A s ω) s ω -
                      ucbWidth A (c * ↑σ2)
                        (A s ω) s ω ≤
                    ∫ (x : ℝ),
                      id x ∂ν (A s ω)}
          1 s =
      0