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

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   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_le_plus.ma".
  16 include "ground/arith/pnat_lt.ma".
  17
  18 (* STRICT ORDER FOR POSITIVE INTEGERS ***************************************)
  19
  20 (* Constructions with pplus *************************************************)
  21
  22 lemma plt_plus_bi_dx (p) (q1) (q2): q1 < q2 → q1 + p < q2 + p.
  23 #p #q1 #q2 #H
  24 @plt_i >pplus_succ_sn /2 width=1 by ple_plus_bi_dx/
  25 qed.
  26
  27 lemma plt_plus_bi_sn (p) (q1) (q2): q1 < q2 → p + q1 < p + q2.
  28 #p #q1 #q2 #H
  29 @plt_i >pplus_succ_dx /2 width=1 by ple_plus_bi_sn/
  30 qed.
  31
  32 lemma plt_plus_dx_dx_refl (p) (q): p < p + q.
  33 /2 width=1 by ple_plus_bi_sn/ qed.
  34
  35 lemma plt_plus_dx_sn_refl (p) (q): p < q + p.
  36 /2 width=1 by ple_plus_bi_dx/ qed.
  37
  38 lemma plt_plus_dx_sn (r) (p) (q): q ≤ p → q < r + p.
  39 /2 width=3 by ple_plt_trans/ qed.
  40
  41 lemma plt_plus_dx_dx (r) (p) (q): q ≤ p → q < p + r.
  42 /2 width=3 by ple_plt_trans/ qed.