(** ** Addition *)
(** Compatibility of addition wrt to order *)
Notation Zplus_le_lt_compat := Z.add_le_lt_mono (only parsing).
Notation Zplus_le_compat := Z.add_le_mono (only parsing).
Notation Zplus_lt_compat := Z.add_lt_mono (only parsing).

(* Specifications: *)
Lemma Zplus_gt_compat_l n m p : n > m -> p + n > p + m.
Lemma Zplus_gt_compat_r n m p : n > m -> n + p > m + p.
Lemma Zplus_le_compat_l n m p : n <= m -> p + n <= p + m.
Lemma Zplus_le_compat_r n m p : n <= m -> n + p <= m + p.
Lemma Zplus_lt_compat_l n m p : n < m -> p + n < p + m.
Lemma Zplus_lt_compat_r n m p : n < m -> n + p < m + p.
Notation Zplus_le_0_compat := Z.add_nonneg_nonneg (only parsing).