import Mathlib
import OAI.Computability.VertexCover.Repetition.SelectedMarginal

namespace OAI

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namespace UniqueGames.Foundations.Repetition
open scoped BigOperators
open Games Information
noncomputable section

theorem isProbability_comp_equiv {A B : Type*} [Fintype A] [Fintype B]
    (e : A ≃ B) (p : B → ℝ) (hp : IsProbability p) :
    IsProbability (fun a => p (e a)) := by
  constructor
  · intro a
    exact hp.1 _
  · rw [e.sum_comp]
    exact hp.2

theorem partialRevealMarginal_isProbability
    {I T X Y : Type*} [Fintype I] [DecidableEq I] [Fintype T]
    [Fintype X] [Fintype Y] [DecidableEq X] [DecidableEq Y]
    (μ : FiniteDistribution (X × Y)) (j : I)
    (likelihood : T → (I → X × Y) → ℝ)
    (hp : IsProbability (fullRevealMarginal μ j likelihood)) :
    IsProbability (partialRevealMarginal μ j likelihood) := by
  apply (maskedJoint_isProbability_iff _).mp
  have h := isProbability_comp_equiv (revealSplitEquiv (T := T) j).symm _ hp
  have he : (fun z => fullRevealMarginal μ j likelihood
      ((revealSplitEquiv (T := T) j).symm z)) =
      maskedJoint (partialRevealMarginal μ j likelihood) := by
    funext z
    exact fullRevealMarginal_merge μ j likelihood z.1.1.1 z.1.1.2 z.1.2 z.2
  rw [he] at h
  exact h

variable {Q₁ Q₂ A₁ A₂ : Type*}
  [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
  [DecidableEq Q₁] [DecidableEq Q₂] {n : Nat}

abbrev SelectedCommonData (selected : Finset (Fin n))
    (j : {i : Fin n // i ∉ selected}) :=
  ((selected → Q₁ × Q₂) × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) ×
    ({i : {i : Fin n // i ∉ selected} // i ≠ j} → Q₁ ⊕ Q₂)

theorem selected_fullReveal_isProbability (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) :
    IsProbability (fullRevealMarginal G.questions j
      (selectedOutsideLikelihood G strategy selected)) := by
  have h := isProbability_comp_equiv
    (selectedObservationEquiv (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected).symm
    _ (selectedRawMarginal_isProbability G strategy selected positive j)
  have he : (fun z => selectedRawMarginal G strategy selected j
      ((selectedObservationEquiv (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected).symm z)) =
      fullRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected) := by
    funext z
    exact selectedRawMarginal_fullReveal G strategy selected j z
  rw [he] at h
  exact h

def selectedCommonLaw (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected}) :
    FiniteDistribution (SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂)
      (A₁ := A₁) (A₂ := A₂) selected j × (Q₁ × Q₂)) :=
  toGameLaw (partialRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected))
    (partialRevealMarginal_isProbability G.questions j _
      (selected_fullReveal_isProbability G strategy selected positive j))

end
end UniqueGames.Foundations.Repetition
end


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end OAI