intros.
apply Zgt_trans with (m := (n + p)%Z).
rewrite Zplus_comm.
rewrite Zplus_comm with (n := n).
apply Zplus_gt_compat_l.
assumption.
apply Zplus_gt_compat_l.
assumption.
Qed.

Lemma Zlt_le_eq_S :
forall x y : Z, (x < y)%Z -> (y <= x + 1)%Z -> y = (x + 1)%Z.