Fixpoint be_x_free (x : BDDvar) (be : bool_expr) {struct be} : bool :=
  match be with
  | Zero => false
  | One => false
  | Var y => Neqb x y
  | Neg be1 => be_x_free x be1
  | Or be1 be2 => be_x_free x be1 || be_x_free x be2
  | ANd be1 be2 => be_x_free x be1 || be_x_free x be2
  | Impl be1 be2 => be_x_free x be1 || be_x_free x be2
  | Iff be1 be2 => be_x_free x be1 || be_x_free x be2
  end.

Lemma exl_x_free : forall (lx : list ad) (be : bool_expr) (x : BDDvar), be_x_free x (exl be lx) = true -> be_x_free x be = true /\ ~ In x lx.