import Mathlib
import OAI.Computability.VertexCover.Encoding.BinaryFormula

namespace OAI

                                                                                        

namespace UniqueGames.Foundations.PCP

structure ConstraintGraph (V E A : Type*) where
  reverse : E ≃ E
  reverse_involutive : Function.Involutive reverse
  tail : E → V
  accepts : E → A → A → Bool
  reverse_accepts : ∀ e a b, accepts (reverse e) b a = accepts e a b

namespace ConstraintGraph

variable {V E A : Type*}

def head (G : ConstraintGraph V E A) (e : E) : V := G.tail (G.reverse e)

@[simp] theorem head_reverse (G : ConstraintGraph V E A) (e : E) :
    G.head (G.reverse e) = G.tail e := by
  simp only [head, G.reverse_involutive e]

def edgeSatisfied (G : ConstraintGraph V E A) (labeling : V → A) (e : E) : Bool :=
  G.accepts e (labeling (G.tail e)) (labeling (G.head e))

@[simp] theorem edgeSatisfied_reverse (G : ConstraintGraph V E A)
    (labeling : V → A) (e : E) :
    G.edgeSatisfied labeling (G.reverse e) = G.edgeSatisfied labeling e := by
  simpa only [edgeSatisfied, head, G.reverse_involutive e] using
    G.reverse_accepts e (labeling (G.tail e)) (labeling (G.tail (G.reverse e)))

def Satisfiable (G : ConstraintGraph V E A) : Prop :=
  ∃ labeling : V → A, ∀ e, G.edgeSatisfied labeling e = true

def rejectedDarts [Fintype E] (G : ConstraintGraph V E A) (labeling : V → A) :
    Finset E := Finset.univ.filter (fun e => G.edgeSatisfied labeling e = false)

def rejectionCount [Fintype E] (G : ConstraintGraph V E A) (labeling : V → A) : Nat :=
  (G.rejectedDarts labeling).card

theorem mem_rejectedDarts [Fintype E] (G : ConstraintGraph V E A)
    (labeling : V → A) (e : E) :
    e ∈ G.rejectedDarts labeling ↔ G.edgeSatisfied labeling e = false := by
  simp [rejectedDarts]

theorem rejectionCount_le [Fintype E] (G : ConstraintGraph V E A)
    (labeling : V → A) : G.rejectionCount labeling ≤ Fintype.card E := by
  exact Finset.card_le_card (Finset.filter_subset _ _)

theorem rejected_exists_of_not_satisfiable (G : ConstraintGraph V E A)
    (unsat : ¬ G.Satisfiable) (labeling : V → A) :
    ∃ e, G.edgeSatisfied labeling e = false := by
  classical
  by_contra none
  apply unsat
  refine ⟨labeling, fun e => ?_⟩
  have notFalse : G.edgeSatisfied labeling e ≠ false := by
    intro h
    exact none ⟨e, h⟩
  cases h : G.edgeSatisfied labeling e <;> simp_all

theorem rejectionCount_positive [Fintype E] (G : ConstraintGraph V E A)
    (unsat : ¬ G.Satisfiable) (labeling : V → A) :
    1 ≤ G.rejectionCount labeling := by
  obtain ⟨e, he⟩ := G.rejected_exists_of_not_satisfiable unsat labeling
  exact Finset.one_le_card.mpr ⟨e, (G.mem_rejectedDarts labeling e).mpr he⟩

end ConstraintGraph

end UniqueGames.Foundations.PCP

end OAI