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

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 include "ground/arith/nat_le_plus.ma".
  16 include "ground/arith/nat_lt.ma".
  17
  18 (* NON-NEGATIVE INTEGERS ****************************************************)
  19
  20 (* Basic constructions with plus ********************************************)
  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 (*** lt_plus_Sn_r *) (**)
  35 lemma lt_plus_Sn_r: ∀a,x,n. a < a + x + ↑n.
  36 /2 width=1/ qed-.
  37
  38 (* Basic inversions with plus ***********************************************)
  39
  40 (*** lt_plus_to_lt_l *)
  41 lemma nlt_inv_plus_bi_dx (m) (n1) (n2): n1 + m < n2 + m → n1 < n2.
  42 /2 width=2 by nle_inv_plus_bi_dx/ qed-.
  43
  44 (*** lt_plus_to_lt_r *)
  45 lemma nlt_inv_plus_bi_sn (m) (n1) (n2): m + n1 < m + n2 → n1 < n2.
  46 /2 width=2 by nle_inv_plus_bi_sn/ qed-.