import Mathlib.MeasureTheory.Order.Lattice
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

/-! # Standalone extraction for `Learning.Algorithm.p0_comap`
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 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 {𝓞 𝓐 𝓨 Ω : Type*} {m𝓞 : MeasurableSpace 𝓞} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨} {mΩ : MeasurableSpace Ω}

/-- One round of interaction: an observation, then an action, then a feedback. -/
abbrev Round (𝓞 𝓐 𝓨 : Type*) := 𝓞 × 𝓐 × 𝓨

/-- History of `n` complete rounds; `n = 0` is the empty history. -/
abbrev Hist (𝓞 𝓐 𝓨 : Type*) (n : ℕ) := Fin n → Round 𝓞 𝓐 𝓨

/-- A stochastic, sequential algorithm.
At each round, it sees an observation in `𝓞`, then takes an action in `𝓐`, and finally receives
feedback in `𝓨`. The action is a random function of the past rounds and the current observation. -/
@[ext]
structure Algorithm (𝓞 𝓐 𝓨 : Type*) [MeasurableSpace 𝓞] [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
    where
  /-- Law of the action of round `n` given the past rounds and the current observation. -/
  policy : (n : ℕ) → Kernel (Hist 𝓞 𝓐 𝓨 n × 𝓞) 𝓐
  /-- The policy is a Markov kernel. -/
  [isMarkovKernel_policy : ∀ n, IsMarkovKernel (policy n)]

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

/-- Distribution of the first action given the first observation: the policy at time `0` applied to
the empty history. -/
noncomputable def Algorithm.p0 (alg : Algorithm 𝓞 𝓐 𝓨) : Kernel 𝓞 𝓐 :=
  (alg.policy 0).sectR default

instance (alg : Algorithm 𝓞 𝓐 𝓨) : IsMarkovKernel (Algorithm.p0 alg) := sorry

end Learning
end

-- ═══ SequentialLearning.Comap ═══
section
open MeasureTheory ProbabilityTheory
namespace Learning
variable {𝓞 𝓞' 𝓞'' 𝓐 𝓐' 𝓐'' 𝓨 𝓨' 𝓨'' Ω : Type*} {m𝓞 : MeasurableSpace 𝓞} {m𝓞' : MeasurableSpace 𝓞'} {m𝓞'' : MeasurableSpace 𝓞''} {m𝓐 : MeasurableSpace 𝓐} {m𝓐' : MeasurableSpace 𝓐'} {m𝓐'' : MeasurableSpace 𝓐''} {m𝓨 : MeasurableSpace 𝓨} {m𝓨' : MeasurableSpace 𝓨'} {m𝓨'' : MeasurableSpace 𝓨''} {mΩ : MeasurableSpace Ω} {fo : 𝓞 → 𝓞'} {fa : 𝓐 → 𝓐'} {fy : 𝓨 → 𝓨'} {go : 𝓞' → 𝓞''} {ga : 𝓐' → 𝓐''} {gy : 𝓨' → 𝓨''}
section Comap

/-- The algorithm with observations in `𝓞'` and feedbacks in `𝓨'` obtained from
`alg : Algorithm 𝓞 𝓐 𝓨` by transforming the pair (past rounds, current observation) by `F n` at
each round `n` before applying the policy of `alg`.

This is the primitive transport operation on algorithms: `Algorithm.comapObs` and
`Algorithm.comapFeedback` are the special cases in which `F n` is a round-wise map of the
observation and of the feedback. -/
def Algorithm.comap (alg : Algorithm 𝓞 𝓐 𝓨)
    (F : (n : ℕ) → Hist 𝓞' 𝓐 𝓨' n × 𝓞' → Hist 𝓞 𝓐 𝓨 n × 𝓞) (hF : ∀ n, Measurable (F n)) :
    Algorithm 𝓞' 𝓐 𝓨' where
  policy n := (alg.policy n).comap (F n) (hF n)

@[simp]
lemma Algorithm.p0_comap (alg : Algorithm 𝓞 𝓐 𝓨)
    {F : (n : ℕ) → Hist 𝓞' 𝓐 𝓨' n × 𝓞' → Hist 𝓞 𝓐 𝓨 n × 𝓞} (hF : ∀ n, Measurable (F n)) :
    (alg.comap F hF).p0
      = alg.p0.comap (fun o ↦ (F 0 (default, o)).2) (((hF 0).comp measurable_prodMk_left).snd) := sorry

end Comap
end Learning
end