import Mathlib
import OAI.Computability.VertexCover.Repetition.SelectedInformation

namespace OAI

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namespace UniqueGames.Foundations.Repetition
open Games
noncomputable section
variable {Q₁ Q₂ A₁ A₂ : Type*}
  [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
  [DecidableEq Q₁] [DecidableEq Q₂] {n : Nat}

def selectedLabelAccepts (G : Game Q₁ Q₂ A₁ A₂) (selected : Finset (Fin n))
    (fixed : selected → Q₁ × Q₂)
    (label : SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) : Bool := by
  classical
  exact decide (∀ i : selected, G.accepts (fixed i).1 (fixed i).2 (label.1 i) (label.2 i) = true)

def selectedLocalAnswers {Q A : Type*} (strategy : (Fin n → Q) → (Fin n → A))
    (selected : Finset (Fin n)) (fixed : selected → Q)
    (remaining : {i : Fin n // i ∉ selected} → Q) : selected → A :=
  fun i => strategy (mergeCoordinates selected fixed remaining) i.1

def selectedLocalTest {Q A : Type*} (strategy : (Fin n → Q) → (Fin n → A))
    (selected : Finset (Fin n)) (fixed : selected → Q) (label : selected → A)
    (remaining : {i : Fin n // i ∉ selected} → Q) : ℝ := by
  classical
  exact if selectedLocalAnswers strategy selected fixed remaining = label then 1 else 0

theorem selectedLocalTest_nonnegative {Q A : Type*} (strategy : (Fin n → Q) → (Fin n → A))
    (selected : Finset (Fin n)) (fixed : selected → Q) (label : selected → A)
    (remaining : {i : Fin n // i ∉ selected} → Q) :
    0 ≤ selectedLocalTest strategy selected fixed label remaining := by
  classical
  unfold selectedLocalTest
  split <;> norm_num

omit [Fintype Q₁] [Fintype Q₂] [DecidableEq Q₁] [DecidableEq Q₂] in
theorem selectedQuestionTuple_left (selected : Finset (Fin n))
    (fixed : selected → Q₁ × Q₂) (remaining : {i : Fin n // i ∉ selected} → Q₁ × Q₂) :
    (selectedQuestionTuple selected fixed remaining).1 =
      mergeCoordinates selected (fun i => (fixed i).1) (fun i => (remaining i).1) := by
  funext i
  simp only [selectedQuestionTuple, mergeCoordinates]
  split <;> rfl

omit [Fintype Q₁] [Fintype Q₂] [DecidableEq Q₁] [DecidableEq Q₂] in
theorem selectedQuestionTuple_right (selected : Finset (Fin n))
    (fixed : selected → Q₁ × Q₂) (remaining : {i : Fin n // i ∉ selected} → Q₁ × Q₂) :
    (selectedQuestionTuple selected fixed remaining).2 =
      mergeCoordinates selected (fun i => (fixed i).2) (fun i => (remaining i).2) := by
  funext i
  simp only [selectedQuestionTuple, mergeCoordinates]
  split <;> rfl

omit [DecidableEq Q₁] [DecidableEq Q₂] in
theorem selectedWins_of_answerLabel (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (fixed : selected → Q₁ × Q₂)
    (label : SelectedLabels (A₁ := A₁) (A₂ := A₂) selected)
    (remaining : {i : Fin n // i ∉ selected} → Q₁ × Q₂)
    (hlabel : selectedAnswerLabel strategy selected (selectedQuestionTuple selected fixed remaining) = label) :
    G.selectedWins strategy selected (selectedQuestionTuple selected fixed remaining) =
      selectedLabelAccepts G selected fixed label := by
  classical
  have hleft (i : selected) := congrArg (fun l => l.1 i) hlabel
  have hright (i : selected) := congrArg (fun l => l.2 i) hlabel
  apply Bool.eq_iff_iff.mpr
  simp only [Game.selectedWins, selectedLabelAccepts, decide_eq_true_eq]
  constructor
  · intro h i
    have hi := h i.1 i.property
    simpa [Game.coordinateWin, selectedQuestionTuple, mergeCoordinates, i.property,
      ← hleft i, ← hright i, selectedAnswerLabel] using hi
  · intro h i hi
    have hv := h ⟨i,hi⟩
    simpa [Game.coordinateWin, selectedQuestionTuple, mergeCoordinates, hi,
      ← hleft ⟨i,hi⟩, ← hright ⟨i,hi⟩, selectedAnswerLabel] using hv

omit [DecidableEq Q₁] [DecidableEq Q₂] in
theorem selectedLikelihood_factorization (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (fixed : selected → Q₁ × Q₂)
    (label : SelectedLabels (A₁ := A₁) (A₂ := A₂) selected)
    (remaining : {i : Fin n // i ∉ selected} → Q₁ × Q₂) :
    selectedLikelihood G strategy selected fixed label remaining =
      (if selectedLabelAccepts G selected fixed label then 1 else 0) *
        selectedLocalTest strategy.1 selected (fun i => (fixed i).1) label.1 (fun i => (remaining i).1) *
        selectedLocalTest strategy.2 selected (fun i => (fixed i).2) label.2 (fun i => (remaining i).2) := by
  classical
  have he : selectedAnswerLabel strategy selected (selectedQuestionTuple selected fixed remaining) =
      (selectedLocalAnswers strategy.1 selected (fun i => (fixed i).1) (fun i => (remaining i).1),
       selectedLocalAnswers strategy.2 selected (fun i => (fixed i).2) (fun i => (remaining i).2)) := by
    simp only [selectedAnswerLabel,
      selectedQuestionTuple_left, selectedQuestionTuple_right]
    rfl
  by_cases hl : selectedLocalAnswers strategy.1 selected (fun i => (fixed i).1)
      (fun i => (remaining i).1) = label.1
  · by_cases hr : selectedLocalAnswers strategy.2 selected (fun i => (fixed i).2)
        (fun i => (remaining i).2) = label.2
    · have hlabel : selectedAnswerLabel strategy selected (selectedQuestionTuple selected fixed remaining) = label := by
        rw [he, hl, hr]
      simp only [selectedLikelihood, ite_eq_left hlabel,
        selectedWins_of_answerLabel G strategy selected fixed label remaining hlabel,
        selectedLocalTest, ite_eq_left hl, ite_eq_left hr, mul_one]
    · have hn : selectedAnswerLabel strategy selected (selectedQuestionTuple selected fixed remaining) ≠ label := by
        intro h
        rw [he] at h
        exact hr (congrArg Prod.snd h)
      simp [selectedLikelihood, hn, selectedLocalTest, hl, hr]
  · have hn : selectedAnswerLabel strategy selected (selectedQuestionTuple selected fixed remaining) ≠ label := by
      intro h
      rw [he] at h
      exact hl (congrArg Prod.fst h)
    simp [selectedLikelihood, hn, selectedLocalTest, hl]

end
end UniqueGames.Foundations.Repetition
end


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end OAI