import Mathlib.MeasureTheory.Order.Lattice
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.SumIntegralComparisons
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.IonescuTulcea.Traj
import Mathlib.Probability.Process.FiniteDimensionalLaws
import Mathlib.Probability.HasCondDistrib
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
import Mathlib.MeasureTheory.Measure.SubFinite
import Mathlib.Probability.Kernel.RadonNikodym
import Mathlib.MeasureTheory.MeasurableSpace.Embedding
import Mathlib.Order.Restriction
import Mathlib.Probability.Kernel.IonescuTulcea.Maps
import Mathlib.Analysis.Normed.Ring.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.Probability.Kernel.Composition.MapComap
import Mathlib.Order.CompletePartialOrder
import Mathlib.Probability.Martingale.BorelCantelli
import Mathlib.CategoryTheory.Countable
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.Probability.Kernel.Representation
import Mathlib.Probability.IdentDistrib
import Mathlib.Probability.Independence.InfinitePi
import Mathlib.MeasureTheory.Function.FactorsThrough
import Mathlib.Probability.Moments.SubGaussian

/-! # Standalone extraction for `Bandits.UCB.constSum_le`
Definitions are copied verbatim; theorem proofs are replaced by `sorry`.
Auto-generated by ChallengeGen. -/

set_option quotPrecheck false

-- Namespace stubs (so later `open`s resolve).
namespace Finset
end Finset
namespace MeasureTheory
end MeasureTheory
namespace ProbabilityTheory
end ProbabilityTheory
namespace Learning
end Learning
namespace ENNReal
end ENNReal
namespace Bandits
end Bandits
namespace Bandits.UCB
end Bandits.UCB

-- ═══ ForMathlib.MeasureTheory.Order.Lattice ═══
section
open Finset
variable {α δ : Type*} [MeasurableSpace δ] [SemilatticeInf α] {m : MeasurableSpace α} [MeasurableInf₂ α]
attribute [to_dual existing] MeasurableInf₂
end

-- ═══ Online.Bandit.Algorithms.Regret.UCB ═══
section
open MeasureTheory ProbabilityTheory Filter Real Finset Learning
open scoped ENNReal NNReal
namespace Bandits
namespace UCB
variable {K : ℕ} [NeZero K] {c : ℝ} {ν : Kernel (Fin K) ℝ} [IsMarkovKernel ν] {Ω : Type*} {mΩ : MeasurableSpace Ω} {P : Measure Ω} [IsProbabilityMeasure P] {O : ℕ → Ω → Unit} {A : ℕ → Ω → Fin K} {R : ℕ → Ω → ℝ} {σ2 : ℝ≥0} {n : ℕ} {ω : Ω}

/-- A sum that appears in the UCB regret upper bound. For `c > 2` it is bounded uniformly in `n`,
see `constSum_le`. -/
noncomputable
def constSum (c : ℝ) (n : ℕ) : ℝ := ∑ s ∈ range n, 1 / ((s : ℝ) + 1) ^ (c - 1)

/-- For `c > 2`, the sum `constSum c n` is at most `1 + 1 / (c - 2)`, uniformly in `n`. -/
lemma constSum_le {c : ℝ} (hc : 2 < c) (n : ℕ) : constSum c n ≤ 1 + 1 / (c - 2) := sorry

end UCB
end Bandits
end