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.CompProdEqIff
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Probability.Kernel.Composition.MeasureCompProd

/-! # Standalone extraction for `Learning.IsAlgEnvSeq.absolutelyContinuous_map_history`
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
namespace Learning.Algorithm
end Learning.Algorithm
namespace Learning.IsAlgEnvSeq
end Learning.IsAlgEnvSeq

-- ═══ 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]

/-- 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]

section IsAlgEnvSeq
variable {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨} {alg : Algorithm 𝓐 𝓨} {env : Environment 𝓐 𝓨} {P : Measure Ω} [IsFiniteMeasure P] {N : ℕ}

/-- History of the algorithm-environment sequence up to time `n`. -/
def history (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) : Iic n → 𝓐 × 𝓨 :=
  fun i ↦ (A i ω, Y i ω)

/-- An algorithm-environment sequence: a sequence of actions and feedbacks generated
by an algorithm interacting with an environment. -/
structure IsAlgEnvSeq
    (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
    (P : Measure Ω) [IsFiniteMeasure P] : Prop where
  /-- The action sequence is measurable. -/
  measurable_action n : Measurable (A n) := sorry
  /-- The feedback sequence is measurable. -/
  measurable_feedback n : Measurable (Y n) := sorry
  /-- The first action has the correct law. -/
  hasLaw_action_zero : HasLaw (fun ω ↦ (A 0 ω)) alg.p0 P
  /-- The first feedback has the correct conditional distribution. -/
  hasCondDistrib_feedback_zero : HasCondDistrib (Y 0) (A 0) env.ν0 P
  /-- The next action has the correct conditional distribution given the history. -/
  hasCondDistrib_action n :
    HasCondDistrib (A (n + 1)) (history A Y n) (alg.policy n) P
  /-- The next feedback has the correct conditional distribution given the history and
  next action. -/
  hasCondDistrib_feedback n :
    HasCondDistrib (Y (n + 1)) (fun ω ↦ (history A Y n ω, A (n + 1) ω))
      (env.feedback n) P

end IsAlgEnvSeq
end Learning
end

-- ═══ SequentialLearning.AlgorithmDensity ═══
section
open MeasureTheory ProbabilityTheory Finset
open scoped ENNReal
namespace Learning
variable {𝓐 𝓨 : Type*} [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
namespace Algorithm

/-- For every time and history, the distribution over actions according to `alg` is absolutely
continuous with respect to the distribution over actions according to `alg₀`. -/
structure AbsolutelyContinuous (alg alg₀ : Algorithm 𝓐 𝓨) : Prop where
  p0 : alg.p0 ≪ alg₀.p0
  policy n h : alg.policy n h ≪ alg₀.policy n h

@[inherit_doc AbsolutelyContinuous]
scoped notation:50 alg " ≪ₐ " alg₀ => AbsolutelyContinuous alg alg₀

end Algorithm
open scoped Algorithm
namespace IsAlgEnvSeq
variable {Ω : Type*} [MeasurableSpace Ω]
variable {alg : Algorithm 𝓐 𝓨} {env : Environment 𝓐 𝓨}
variable {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨}
variable {P : Measure Ω} [IsFiniteMeasure P]
variable {Ω₀ : Type*} [MeasurableSpace Ω₀]
variable {alg₀ : Algorithm 𝓐 𝓨}
variable {A₀ : ℕ → Ω₀ → 𝓐} {Y₀ : ℕ → Ω₀ → 𝓨}
variable {P₀ : Measure Ω₀} [IsProbabilityMeasure P₀]

lemma absolutelyContinuous_map_history (h : IsAlgEnvSeq A Y alg env P)
    (h₀ : IsAlgEnvSeq A₀ Y₀ alg₀ env P₀) (hc : alg ≪ₐ alg₀) (n : ℕ) :
    P.map (history A Y n) ≪ P₀.map (history A₀ Y₀ n) := sorry

variable [MeasurableSpace.CountablyGenerated 𝓐]
end IsAlgEnvSeq
end Learning
end