Bandits.ETC.regret_le.{u_1} {K : ℕ} {hK : 0 < K} {m : ℕ}
  {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 (etcAlgorithm hK m)
      (Learning.stationaryEnv ν) P)
  (hν :
    ∀ (a : Fin K),
      ProbabilityTheory.HasSubgaussianMGF
        (fun x => x - ∫ (x : ℝ), id x ∂ν a) σ2 (ν a))
  (hm : m ≠ 0) (n : ℕ) (hn : K * m ≤ n) :
  ∫ (x : Ω), regret ν A n x ∂P ≤
    ∑ a,
      gap ν a *
        (↑m + (↑n - ↑K * ↑m) * Real.exp (-↑m * gap ν a ^ 2 / (4 * ↑σ2)))
Bandits.ETC.regret_le.{u_1} {K : ℕ}
  {hK : 0 < K} {m : ℕ}
  {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
      (etcAlgorithm hK m)
      (Learning.stationaryEnv ν) P)
  (hν :
    ∀ (a : Fin K),
      ProbabilityTheory.HasSubgaussianMGF
        (fun x =>
          x - ∫ (x : ℝ), id x ∂ν a)
        σ2 (ν a))
  (hm : m ≠ 0) (n : ℕ) (hn : K * m ≤ n) :
  ∫ (x : Ω), regret ν A n x ∂P ≤
    ∑ a,
      gap ν a *
        (↑m +
          (↑n - ↑K * ↑m) *
            Real.exp
              (-↑m * gap ν a ^ 2 /
                (4 * ↑σ2)))