Definition Rbar_minus (x y : Rbar) := Rbar_plus x (Rbar_opp y).Definition is_Rbar_minus (x y z : Rbar) : Prop :=
  is_Rbar_plus x (Rbar_opp y) z.Definition ex_Rbar_minus (x y : Rbar) : Prop :=
  ex_Rbar_plus x (Rbar_opp y).(** *** Multiplicative operators *)

Definition Rbar_inv (x : Rbar) : Rbar :=
  match x with
    | Finite x => Finite (/x)
    | _ => Finite 0
  end.Definition Rbar_mult' (x y : Rbar) :=
  match x with
    | Finite x => match y with
      | Finite y => Some (Finite (x * y))
      | p_infty => match (Rle_dec 0 x) with
        | left H => match Rle_lt_or_eq_dec _ _ H with left _ => Some p_infty | right _ => None end
        | right _ => Some m_infty
      end
      | m_infty => match (Rle_dec 0 x) with
        | left H => match Rle_lt_or_eq_dec _ _ H with left _ => Some m_infty | right _ => None end
        | right _ => Some p_infty
      end
    end
    | p_infty => match y with
      | Finite y => match (Rle_dec 0 y) with
        | left H => match Rle_lt_or_eq_dec _ _ H with left _ => Some p_infty | right _ => None end
        | right _ => Some m_infty
      end
      | p_infty => Some p_infty
      | m_infty => Some m_infty
    end
    | m_infty => match y with
      | Finite y => match (Rle_dec 0 y) with
        | left H => match Rle_lt_or_eq_dec _ _ H with left _ => Some m_infty | right _ => None end
        | right _ => Some p_infty
      end
      | p_infty => Some m_infty
      | m_infty => Some p_infty
    end
  end.Definition Rbar_mult (x y : Rbar) :=
  match Rbar_mult' x y with Some z => z | None => Finite 0 end.Definition is_Rbar_mult (x y z : Rbar) : Prop :=
  Rbar_mult' x y = Some z.Definition ex_Rbar_mult (x y : Rbar) : Prop :=
  match x with
    | Finite x => match y with
      | Finite y => True
      | p_infty => x <> 0
      | m_infty => x <> 0
    end
    | p_infty => match y with
      | Finite y => y <> 0
      | p_infty => True
      | m_infty => True
    end
    | m_infty => match y with
      | Finite y => y <> 0
      | p_infty => True
      | m_infty => True
    end
  end.Definition Rbar_mult_pos (x : Rbar) (y : posreal) :=
  match x with
    | Finite x => Finite (x*y)
    | _ => x
  end.

Lemma is_Rbar_mult_unique (x y z : Rbar) : is_Rbar_mult x y z -> Rbar_mult x y = z.