(** ** Order *)

Definition Rbar_lt (x y : Rbar) : Prop :=
  match x,y with
    | p_infty, _ | _, m_infty => False
    | m_infty, _ | _, p_infty => True
    | Finite x, Finite y => Rlt x y
  end.Definition Rbar_le (x y : Rbar) : Prop :=
  match x,y with
    | m_infty, _ | _, p_infty => True
    | p_infty, _ | _, m_infty => False
    | Finite x, Finite y => Rle x y
  end.(** ** Operations *)
(** *** Additive operators *)

Definition Rbar_opp (x : Rbar) :=
  match x with
    | Finite x => Finite (-x)
    | p_infty => m_infty
    | m_infty => p_infty
  end.Definition Rbar_plus' (x y : Rbar) :=
  match x,y with
    | p_infty, m_infty | m_infty, p_infty => None
    | p_infty, _ | _, p_infty => Some p_infty
    | m_infty, _ | _, m_infty => Some m_infty
    | Finite x', Finite y' => Some (Finite (x' + y'))
  end.Definition Rbar_plus (x y : Rbar) :=
  match Rbar_plus' x y with Some z => z | None => Finite 0 end.Definition is_Rbar_plus (x y z : Rbar) : Prop :=
  Rbar_plus' x y = Some z.Definition ex_Rbar_plus (x y : Rbar) : Prop :=
  match Rbar_plus' x y with Some _ => True | None => False end.

Lemma Rbar_plus_correct (x y : Rbar) : ex_Rbar_plus x y -> is_Rbar_plus x y (Rbar_plus x y).