(** Simplification of multiplication by a positive wrt to being positive *)

(* Specifications: *)
Lemma Zmult_gt_0_lt_reg_r n m p : p > 0 -> n * p < m * p -> n < m.
Lemma Zmult_lt_reg_r n m p : 0 < p -> n * p < m * p -> n < m.
Lemma Zmult_le_reg_r n m p : p > 0 -> n * p <= m * p -> n <= m.
Lemma Zmult_lt_0_le_reg_r n m p : 0 < p -> n * p <= m * p -> n <= m.
Lemma Zmult_ge_reg_r n m p : p > 0 -> n * p >= m * p -> n >= m.
Lemma Zmult_gt_reg_r n m p : p > 0 -> n * p > m * p -> n > m.
Lemma Zmult_lt_compat n m p q :
0 <= n < p -> 0 <= m < q -> n * m < p * q.
Lemma Zmult_lt_compat2 n m p q :
0 < n <= p -> 0 < m < q -> n * m < p * q.