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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   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_plus.ma".
  16 include "ground/arith/nat_lt.ma".
  17
  18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
  19
  20 (* Constructions with nplus *************************************************)
  21
  22 (*** monotonic_lt_plus_l *)
  23 lemma nlt_plus_bi_dx (m) (n1) (n2): n1 < n2 → n1 + m < n2 + m.
  24 #m #n1 #n2 #H
  25 @nlt_i >nplus_succ_sn /2 width=1 by nle_plus_bi_dx/
  26 qed.
  27
  28 (*** monotonic_lt_plus_r *)
  29 lemma nlt_plus_bi_sn (m) (n1) (n2): n1 < n2 → m + n1 < m + n2.
  30 #m #n1 #n2 #H
  31 @nlt_i >nplus_succ_dx /2 width=1 by nle_plus_bi_sn/
  32 qed.
  33
  34 lemma nlt_succ_plus_dx_refl_sn (m) (n): m < ↑(m + n).
  35 /2 width=1/ qed.
  36
  37 (* Inversions with nplus ****************************************************)
  38
  39 (*** lt_plus_to_lt_l *)
  40 lemma nlt_inv_plus_bi_dx (m) (n1) (n2): n1 + m < n2 + m → n1 < n2.
  41 /2 width=2 by nle_inv_plus_bi_dx/ qed-.
  42
  43 (*** lt_plus_to_lt_r *)
  44 lemma nlt_inv_plus_bi_sn (m) (n1) (n2): m + n1 < m + n2 → n1 < n2.
  45 /2 width=2 by nle_inv_plus_bi_sn/ qed-.