import Mathlib.MeasureTheory.Order.Lattice
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

/-! # Standalone extraction for `Learning.trajMeasure`
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 ENNReal
end ENNReal
namespace Learning
end Learning

-- ═══ 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

-- ═══ SequentialLearning.Algorithm ═══
section
open MeasureTheory ProbabilityTheory Filter Real Finset
open scoped ENNReal NNReal
namespace Learning
variable {𝓐 𝓨 Ω : Type*} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨} {mΩ : MeasurableSpace Ω}

/-- A stochastic, sequential algorithm. -/
structure Algorithm (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Policy or sampling rule: distribution of the next action. -/
  policy : (n : ℕ) → Kernel (Iic n → 𝓐 × 𝓨) 𝓐
  /-- The policy is a Markov kernel. -/
  [h_policy : ∀ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure 𝓐
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]

instance (alg : Algorithm 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (alg.policy n) := alg.h_policy n

instance (alg : Algorithm 𝓐 𝓨) : IsProbabilityMeasure alg.p0 := alg.hp0

/-- A stochastic environment. -/
structure Environment (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Distribution of the next observation as function of the past history. -/
  feedback : (n : ℕ) → Kernel ((Iic n → 𝓐 × 𝓨) × 𝓐) 𝓨
  /-- The feedback kernels are Markov kernels. -/
  [h_feedback : ∀ n, IsMarkovKernel (feedback n)]
  /-- Distribution of the first observation given the first action. -/
  ν0 : Kernel 𝓐 𝓨
  /-- The initial observation kernel is a Markov kernel. -/
  [hp0 : IsMarkovKernel ν0]

instance (env : Environment 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (env.feedback n) := env.h_feedback n

instance (env : Environment 𝓐 𝓨) : IsMarkovKernel env.ν0 := env.hp0

/-- Kernel describing the distribution of the next action-feedback pair given the history
up to `n`. -/
noncomputable
def stepKernel (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :
    Kernel (Iic n → 𝓐 × 𝓨) (𝓐 × 𝓨) :=
  alg.policy n ⊗ₖ env.feedback n

instance (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (stepKernel alg env n) := sorry

end Learning
end

-- ═══ SequentialLearning.IonescuTulceaSpace ═══
section
open MeasureTheory ProbabilityTheory Filter Real Finset
open scoped ENNReal NNReal
namespace Learning
variable {𝓐 𝓨 Ω : Type*} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨} {mΩ : MeasurableSpace Ω}

/-- Measure on the sequence of actions and observations generated by the algorithm/environment. -/
noncomputable
def trajMeasure (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) :
    Measure (ℕ → 𝓐 × 𝓨) :=
  Kernel.trajMeasure (alg.p0 ⊗ₘ env.ν0) (stepKernel alg env)

instance (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) : IsProbabilityMeasure (trajMeasure alg env) := sorry

end Learning
end