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
import Mathlib.Probability.Kernel.Basic

/-! # Standalone extraction for `Learning.feedbackFunZero`
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 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]

end Learning
end

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

/-- An environment is deterministic if its initial feedbacks are determined by
measurable functions (and not possibly random kernels). -/
class IsDeterministicEnv (env : Environment 𝓐 𝓨) : Prop where
  exists_f0 : ∃ (f0 : 𝓐 → 𝓨) (hf0 : Measurable f0), env.ν0 = Kernel.deterministic f0 hf0
  exists_f : ∀ n, ∃ (f : ((Iic n → 𝓐 × 𝓨) × 𝓐) → 𝓨) (hf : Measurable f),
    env.feedback n = Kernel.deterministic f hf

/-- The initial feedback function of a deterministic environment. -/
noncomputable
def feedbackFunZero (env : Environment 𝓐 𝓨) [h_det : IsDeterministicEnv env] : 𝓐 → 𝓨 :=
  h_det.exists_f0.choose

variable {nextA : (n : ℕ) → (Iic n → 𝓐 × 𝓨) → 𝓐} {h_next : ∀ n, Measurable (nextA n)} {action0 : 𝓐} {env : Environment 𝓐 𝓨} {f0 : 𝓐 → 𝓨} {hf0 : Measurable f0} {f : (n : ℕ) → ((Iic n → 𝓐 × 𝓨) × 𝓐) → 𝓨} {hf : ∀ n, Measurable (f n)}
end Learning
end