matita/matita/contribs/lambdadelta/ground/arith/ynat_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_le_pred.ma".
  16 include "ground/arith/ynat_pred.ma".
  17 include "ground/arith/ynat_le.ma".
  18
  19 (* ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY ****************************)
  20
  21 (* Constructions with ypred *************************************************)
  22
  23 (*** yle_pred_sn *)
  24 lemma yle_pred_sn_trans (x) (y): x ≤ y → ↓x ≤ y.
  25 #x #y * -x -y
  26 /3 width=3 by yle_inj, nle_trans/
  27 qed.
  28
  29 (*** yle_refl_pred_sn *)
  30 lemma yle_pred_sn_refl (x): ↓x ≤ x.
  31 /2 width=1 by yle_pred_sn_trans, yle_refl/ qed.
  32
  33 (*** yle_pred *)
  34 lemma yle_pred_bi (x) (y): x ≤ y → ↓x ≤ ↓y.
  35 #x #y * -x -y
  36 /3 width=1 by yle_inj, nle_pred_bi/
  37 qed.