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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   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_minus.ma".
  16 include "ground/arith/nat_lt_pred.ma".
  17
  18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
  19
  20 (* Constructions with nminus ************************************************)
  21
  22 (*** monotonic_lt_minus_l *)
  23 lemma nlt_minus_sn_bi (o) (m) (n): o ≤ m → m < n → m - o < n - o.
  24 #o #m #n #Hom #Hmn
  25 lapply (nle_minus_sn_bi … o Hmn) -Hmn
  26 <(nminus_succ_sn … Hom) //
  27 qed.
  28
  29 (*** monotonic_lt_minus_r *)
  30 lemma nlt_minus_dx_bi (o) (m) (n):
  31       m < o -> m < n → o-n < o-m.
  32 #o #m #n #Ho #H
  33 lapply (nle_minus_dx_bi … o H) -H #H
  34 @(le_nlt_trans … H) -n
  35 @nlt_i >(nminus_succ_sn … Ho) //
  36 qed.
  37
  38 (* Destructions with nminus *************************************************)
  39
  40 (*** minus_pred_pred *)
  41 lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m.
  42 #m #n #Hm #Hn
  43 >(nlt_des_gen … Hm) in ⊢ (??%?); -Hm
  44 >(nlt_des_gen … Hn) in ⊢ (??%?); -Hn
  45 //
  46 qed-.
  47
  48 lemma nlt_des_minus_dx (o) (m) (n): m < n - o → o < n.
  49 #o @(nat_ind_succ … o) -o
  50 [ #m #n <nminus_zero_dx
  51   /2 width=3 by le_nlt_trans/
  52 | #o #IH #m #n <nminus_succ_dx_pred_sn #H
  53   /3 width=2 by nlt_inv_pred_dx/
  54 ]
  55 qed-.