(** * Properties of operations *)
(** ** Additive operators *)
(** *** Rbar_opp *)

(* Specifications: *)
Lemma Rbar_opp_involutive (x : Rbar) : (Rbar_opp (Rbar_opp x)) = x.
Lemma Rbar_opp_lt (x y : Rbar) : Rbar_lt (Rbar_opp x) (Rbar_opp y) <-> Rbar_lt y x.
Lemma Rbar_opp_le (x y : Rbar) : Rbar_le (Rbar_opp x) (Rbar_opp y) <-> Rbar_le y x.
Lemma Rbar_opp_eq (x y : Rbar) : (Rbar_opp x) = (Rbar_opp y) <-> x = y.
Lemma Rbar_opp_real (x : Rbar) : real (Rbar_opp x) = - real x.
(** *** Rbar_plus *)
Lemma Rbar_plus'_comm :
forall x y, Rbar_plus' x y = Rbar_plus' y x.
Lemma ex_Rbar_plus_comm :
forall x y,
ex_Rbar_plus x y -> ex_Rbar_plus y x.
Lemma ex_Rbar_plus_opp (x y : Rbar) :
ex_Rbar_plus x y -> ex_Rbar_plus (Rbar_opp x) (Rbar_opp y).
Lemma Rbar_plus_0_r (x : Rbar) : Rbar_plus x (Finite 0) = x.
Lemma Rbar_plus_0_l (x : Rbar) : Rbar_plus (Finite 0) x = x.
Lemma Rbar_plus_comm (x y : Rbar) : Rbar_plus x y = Rbar_plus y x.
Lemma Rbar_plus_lt_compat (a b c d : Rbar) :
Rbar_lt a b -> Rbar_lt c d -> Rbar_lt (Rbar_plus a c) (Rbar_plus b d).
Lemma Rbar_plus_le_compat (a b c d : Rbar) :
Rbar_le a b -> Rbar_le c d -> Rbar_le (Rbar_plus a c) (Rbar_plus b d).
Lemma Rbar_plus_opp (x y : Rbar) :
Rbar_plus (Rbar_opp x) (Rbar_opp y) = Rbar_opp (Rbar_plus x y).