1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "ground/arith/pnat_le_plus.ma".
16 include "ground/arith/pnat_lt.ma".
18 (* STRICT ORDER FOR POSITIVE INTEGERS ***************************************)
20 (* Constructions with pplus *************************************************)
22 lemma plt_plus_bi_dx (p) (q1) (q2): q1 < q2 → q1 + p < q2 + p.
24 @plt_i >pplus_succ_sn /2 width=1 by ple_plus_bi_dx/
27 lemma plt_plus_bi_sn (p) (q1) (q2): q1 < q2 → p + q1 < p + q2.
29 @plt_i >pplus_succ_dx /2 width=1 by ple_plus_bi_sn/
32 lemma plt_plus_dx_dx_refl (p) (q): p < p + q.
33 /2 width=1 by ple_plus_bi_sn/ qed.
35 lemma plt_plus_dx_sn_refl (p) (q): p < q + p.
36 /2 width=1 by ple_plus_bi_dx/ qed.
38 lemma plt_plus_dx_sn (r) (p) (q): q ≤ p → q < r + p.
39 /2 width=3 by ple_plt_trans/ qed.
41 lemma plt_plus_dx_dx (r) (p) (q): q ≤ p → q < p + r.
42 /2 width=3 by ple_plt_trans/ qed.