matita/matita/contribs/lambdadelta/ground/arith/pnat_le_plus.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/arith/pnat_plus.ma".
  16 include "ground/arith/pnat_le.ma".
  17
  18 (* ORDER FOR POSITIVE INTEGERS **********************************************)
  19
  20 (* Constructions with pplus *************************************************)
  21
  22 lemma ple_plus_bi_dx (p) (q1) (q2): q1 ≤ q2 → q1 + p ≤ q2 + p.
  23 #p #q1 #q2 #H elim H -q2 /2 width=3 by ple_trans/
  24 qed.
  25
  26 lemma ple_plus_bi_sn (p) (q1) (q2): q1 ≤ q2 → p + q1 ≤ p + q2.
  27 #p #q1 #q2 #H <pplus_comm <pplus_comm in ⊢ (??%);
  28 /2 width=1 by ple_plus_bi_dx/
  29 qed.