matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.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/nat_pred_succ.ma".
  16 include "ground/arith/nat_le.ma".
  17
  18 (* ORDER FOR NON-NEGATIVE INTEGERS ******************************************)
  19
  20 (* Constructions with npred *************************************************)
  21
  22 (*** le_pred_n *)
  23 lemma nle_pred_sn_refl (m): ↓m ≤ m.
  24 #m elim m -m //
  25 qed.
  26
  27 (*** monotonic_pred *)
  28 lemma nle_pred_bi (m) (n): m ≤ n → ↓m ≤ ↓n.
  29 #m #n #H elim H -n //
  30 /2 width=3 by nle_trans/
  31 qed.
  32
  33 lemma nle_pred_sn (m) (n): m ≤ ↑n → ↓m ≤ n.
  34 #m #n elim m -m //
  35 /2 width=1 by nle_pred_bi/
  36 qed-.
  37
  38 (* Destructions with npred **************************************************)
  39
  40 lemma nle_inv_pred_sn (m) (n): ↓m ≤ n → m ≤ ↑n.
  41 #m #n elim m -m
  42 /2 width=1 by nle_succ_bi/
  43 qed-.