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.Environment.feedback_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 environment.
At each round, an observation is drawn prior to the algorithm taking an action. Then the environment
provides feedback based on the observation and the action. -/
@[ext]
structure Environment (𝓞 𝓐 𝓨 : Type*) [MeasurableSpace 𝓞] [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]
    where
  /-- Law of the observation of round `n` given the past rounds. -/
  obs : (n : ℕ) → Kernel (Hist 𝓞 𝓐 𝓨 n) 𝓞
  /-- Law of the feedback of round `n` given the past rounds, the observation and the action. -/
  feedback : (n : ℕ) → Kernel ((Hist 𝓞 𝓐 𝓨 n × 𝓞) × 𝓐) 𝓨
  /-- The observation kernel is a Markov kernel. -/
  [isMarkovKernel_obs : ∀ n, IsMarkovKernel (obs n)]
  /-- The feedback kernel is a Markov kernel. -/
  [isMarkovKernel_feedback : ∀ n, IsMarkovKernel (feedback n)]

instance (env : Environment 𝓞 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (env.obs n) := env.isMarkovKernel_obs n

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

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
section ComapAction
variable {F : (n : ℕ) → Hist 𝓞 𝓐' 𝓨 n → Hist 𝓞 𝓐 𝓨 n} {f : 𝓐' → 𝓐}

/-- The environment that reads the summary `F n` of the past rounds and reads `f a` when the
algorithm plays `a` in the current round.

This is the primitive transport operation on environments, dual to `Algorithm.comap`:
`Environment.comapAction` is the special case in which `F n` is the round-wise map of the actions.
Only the action can change type, since the observations and the feedbacks are outputs of the
environment; `F n` can nonetheless forget or summarize the past rounds, as an environment that
reads only the last round does. -/
def Environment.comap (env : Environment 𝓞 𝓐 𝓨)
    (F : (n : ℕ) → Hist 𝓞 𝓐' 𝓨 n → Hist 𝓞 𝓐 𝓨 n) (hF : ∀ n, Measurable (F n))
    (f : 𝓐' → 𝓐) (hf : Measurable f) : Environment 𝓞 𝓐' 𝓨 where
  obs n := (env.obs n).comap (F n) (hF n)
  feedback n := (env.feedback n).comap (fun p ↦ ((F n p.1.1, p.1.2), f p.2)) (sorry)

@[simp]
lemma Environment.feedback_comap (env : Environment 𝓞 𝓐 𝓨) (hF : ∀ n, Measurable (F n))
    (hf : Measurable f) (n : ℕ) :
    (env.comap F hF f hf).feedback n
      = (env.feedback n).comap (fun p ↦ ((F n p.1.1, p.1.2), f p.2)) (by fun_prop) := sorry

end ComapAction
end Comap
end Learning
end