(** * Equivalence between inequalities *)

Notation Zle_plus_swap := Z.le_add_le_sub_r (only parsing).Notation Zlt_plus_swap := Z.lt_add_lt_sub_r (only parsing).Notation Zlt_minus_simpl_swap := Z.lt_sub_pos (only parsing).

(* Specifications: *)
Lemma Zeq_plus_swap n m p : n + p = m <-> n = m - p.
Lemma Zlt_0_minus_lt n m : 0 < n - m -> m < n.
Lemma Zle_0_minus_le n m : 0 <= n - m -> m <= n.
Lemma Zle_minus_le_0 n m : m <= n -> 0 <= n - m.