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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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  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_succ_tri.ma".
  16 include "ground/arith/nat_lt.ma".
  17
  18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
  19
  20 (* Destructions with ntri ***************************************************)
  21
  22 (*** tri_lt *)
  23 lemma ntri_lt (A) (a1) (a2) (a3) (n1) (n2):
  24       n1 < n2 → a1 = ntri A n1 n2 a1 a2 a3.
  25 #A #a1 #a2 #a3 #n1 #n2 #H @(nlt_ind_alt … H) -H //
  26 qed-.
  27
  28 (*** tri_gt *)
  29 lemma ntri_gt (A) (a1) (a2) (a3) (n1) (n2):
  30       n2 < n1 → a3 = ntri A n1 n2 a1 a2 a3.
  31 #A #a1 #a2 #a3 #n1 #n2 #H @(nlt_ind_alt … H) -H //
  32 qed-.