matita/matita/contribs/lambdadelta/ground/arith/nat_lt_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/nat_lt.ma".
  17
  18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
  19
  20 (* Constructions with npred *************************************************)
  21
  22 lemma nlt_zero_sn (m): m = ↑↓m → 𝟎 < m.
  23 // qed.
  24
  25 (* Inversions with npred ****************************************************)
  26
  27 (*** S_pred *)
  28 lemma nlt_inv_zero_sn (m): 𝟎 < m → m = ↑↓m.
  29 #m @(nat_ind_succ … m) -m //
  30 #H elim (nlt_inv_refl … H)
  31 qed-.
  32
  33 lemma nlt_inv_pred_dx (m) (n): m < ↓n → ↑m < n.
  34 #m #n #H >(nlt_inv_zero_sn n)
  35 [ /2 width=1 by nlt_succ_bi/
  36 | /3 width=3 by le_nlt_trans, nlt_le_trans/
  37 ]
  38 qed-.