import Mathlib.MeasureTheory.Order.Lattice
import Mathlib.Probability.Independence.Integration
import Mathlib.Probability.Kernel.Representation
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.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.Probability.IdentDistrib
import Mathlib.Probability.Independence.InfinitePi
import Mathlib.MeasureTheory.Function.FactorsThrough
import Mathlib.CategoryTheory.Countable
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.Probability.Moments.SubGaussian

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

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.ArrayModel
end Bandits.ArrayModel

-- ═══ 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.ArrayProbSpace ═══
section
open MeasureTheory ProbabilityTheory Filter Real Finset Learning
open scoped ENNReal NNReal
namespace Bandits
variable {𝓐 R : Type*} {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
section MeasureSpace

/-- Measure of an infinite stream of rewards from each action. -/
noncomputable
def streamMeasure (ν : Kernel 𝓐 R) : Measure (ℕ → 𝓐 → R) :=
  Measure.infinitePi fun _ ↦ Measure.infinitePi ν

end MeasureSpace
end Bandits
end

-- ═══ Online.Bandit.SumRewards ═══
section
open MeasureTheory ProbabilityTheory Finset Learning
open scoped ENNReal NNReal
namespace Bandits
namespace ArrayModel
variable {𝓐 : Type*} {m𝓐 : MeasurableSpace 𝓐} [DecidableEq 𝓐] [Countable 𝓐] [StandardBorelSpace 𝓐] [Nonempty 𝓐] {ν : Kernel 𝓐 ℝ} [IsMarkovKernel ν]
local notation "A" => action alg
local notation "R" => reward alg
local notation "𝔓" => arrayMeasure ν
end ArrayModel
variable {𝓐 Ω Ω' : Type*} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {P : Measure Ω} [IsProbabilityMeasure P] {P' : Measure Ω'} [IsProbabilityMeasure P'] {ν : Kernel 𝓐 ℝ} [IsMarkovKernel ν] {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ} {A₂ : ℕ → Ω' → 𝓐} {R₂ : ℕ → Ω' → ℝ} {ω : Ω} {m n t : ℕ} {a : 𝓐}
variable [Nonempty 𝓐]
section Subgaussian
open Real

omit [DecidableEq 𝓐] [Nonempty 𝓐] in
lemma prob_avg_add_sqrt_log_le {σ2 : ℝ≥0} {c : ℝ}
    (hν : ∀ a, HasSubgaussianMGF (fun x ↦ x - (ν a)[id]) σ2 (ν a)) (hσ2 : σ2 ≠ 0)
    (hc : 0 ≤ c) (a : 𝓐) (n k : ℕ) (hk : k ≠ 0) :
    streamMeasure ν {ω | (∑ m ∈ range k, ω m a) / k + √(2 * c * σ2 * log (n + 1) / k) ≤ (ν a)[id]} ≤
      1 / (n + 1) ^ c := sorry

end Subgaussian
end Bandits
end