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 (* Destructions with npred **************************************************)
  21
  22 lemma nle_inv_pred_sn (m) (n): ↓m ≤ n → m ≤ ↑n.
  23 #m #n @(nat_ind_succ … m) -m
  24 /2 width=1 by nle_succ_bi/
  25 qed-.
  26
  27 (* Constructions with npred *************************************************)
  28
  29 lemma nle_succ_pred_dx_refl (m): m ≤ ↑↓m.
  30 #m @nle_inv_pred_sn // qed.
  31
  32 (*** le_pred_n *)
  33 lemma nle_pred_sn_refl (m): ↓m ≤ m.
  34 #m @(nat_ind_succ … m) -m //
  35 qed.
  36
  37 (*** monotonic_pred *)
  38 lemma nle_pred_bi (m) (n): m ≤ n → ↓m ≤ ↓n.
  39 #m #n #H elim H -n //
  40 /2 width=3 by nle_trans/
  41 qed.
  42
  43 lemma nle_pred_sn (m) (n): m ≤ ↑n → ↓m ≤ n.
  44 #m #n @(nat_ind_succ … m) -m //
  45 /2 width=1 by nle_pred_bi/
  46 qed-.