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.congr_refl`
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

/-- The observation of a round. -/
def Round.obs (r : Round 𝓞 𝓐 𝓨) : 𝓞 := r.1

/-- The action of a round. -/
def Round.action (r : Round 𝓞 𝓐 𝓨) : 𝓐 := r.2.1

/-- The feedback of a round. -/
def Round.feedback (r : Round 𝓞 𝓐 𝓨) : 𝓨 := r.2.2

/-- 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

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 Map

/-- Transport a round along maps of the observation, the action and the feedback. -/
def Round.map (fo : 𝓞 → 𝓞') (fa : 𝓐 → 𝓐') (fy : 𝓨 → 𝓨') (r : Round 𝓞 𝓐 𝓨) : Round 𝓞' 𝓐' 𝓨' :=
  (fo r.obs, fa r.action, fy r.feedback)

/-- Transport a history round-wise. -/
def Hist.map (fo : 𝓞 → 𝓞') (fa : 𝓐 → 𝓐') (fy : 𝓨 → 𝓨') {n : ℕ} (h : Hist 𝓞 𝓐 𝓨 n) :
    Hist 𝓞' 𝓐' 𝓨' n :=
  fun i ↦ Round.map fo fa fy (h i)

@[fun_prop]
lemma Hist.measurable_map (hfo : Measurable fo) (hfa : Measurable fa) (hfy : Measurable fy)
    (n : ℕ) :
    Measurable (Hist.map fo fa fy (n := n)) := sorry

end Map
section Congr

/-- Relabelling of the observations, the actions and the feedbacks of an algorithm along measurable
equivalences. -/
noncomputable def Algorithm.congr (alg : Algorithm 𝓞 𝓐 𝓨) (e𝓞 : 𝓞 ≃ᵐ 𝓞') (e𝓐 : 𝓐 ≃ᵐ 𝓐')
    (e𝓨 : 𝓨 ≃ᵐ 𝓨') : Algorithm 𝓞' 𝓐' 𝓨' where
  policy n := ((alg.policy n).map e𝓐).comap
    (fun p ↦ (Hist.map e𝓞.symm e𝓐.symm e𝓨.symm p.1, e𝓞.symm p.2)) (sorry)
  isMarkovKernel_policy n := by
    have : IsMarkovKernel ((alg.policy n).map e𝓐) := Kernel.IsMarkovKernel.map _ e𝓐.measurable
    infer_instance

@[simp]
lemma Algorithm.congr_refl (alg : Algorithm 𝓞 𝓐 𝓨) :
    alg.congr (.refl 𝓞) (.refl 𝓐) (.refl 𝓨) = alg := sorry

end Congr
end Learning
end