import Mathlib.MeasureTheory.Order.Lattice
import Mathlib.Probability.Kernel.Representation
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.Probability.IdentDistrib
import Mathlib.Probability.Independence.InfinitePi
import Mathlib.MeasureTheory.Function.FactorsThrough

/-! # Standalone extraction for `Bandits.indepFun_snd_apply_prod_streamMeasure_update`
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

-- ═══ 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 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

section StreamMeasure

/-- Under a product measure `μ.prod (streamMeasure ν)`, the entry `(m, a)` of the reward array is
independent of the pair formed by the first coordinate and the reward array in which the entry
`(m, a)` is replaced by a constant `x`. -/
lemma indepFun_snd_apply_prod_streamMeasure_update [DecidableEq 𝓐] {Ω : Type*}
    {mΩ : MeasurableSpace Ω} (μ : Measure Ω) [IsProbabilityMeasure μ] (ν : Kernel 𝓐 𝓡)
    [IsMarkovKernel ν] (m : ℕ) (a : 𝓐) (x : 𝓡) :
    (fun ω : Ω × (ℕ → 𝓐 → 𝓡) ↦ ω.2 m a) ⟂ᵢ[μ.prod (streamMeasure ν)]
      (fun ω ↦ (ω.1, fun i b ↦ if i = m ∧ b = a then x else ω.2 i b)) := sorry

end StreamMeasure
end MeasureSpace
end Bandits
end