matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_minus1.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_lt_minus.ma".
  16 include "ground/arith/ynat_minus1.ma".
  17 include "ground/arith/ynat_lt.ma".
  18
  19 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
  20
  21 (* Constructions with yminus1 ***********************************************)
  22
  23 (*** ylt_to_minus *)
  24 lemma ylt_zero_minus1 (m) (y):
  25       yinj_nat m < y → 𝟎 < y - m.
  26 #m #y @(ynat_split_nat_inf … y) -y //
  27 #n #Hmn <yminus1_inj_sn >yinj_nat_zero >(nminus_refl m)
  28 /4 width=1 by ylt_inj, ylt_inv_inj_bi, nlt_minus_bi_dx/
  29 qed.
  30
  31 (* Inversions with yminus1 **************************************************)
  32
  33 (*** yminus_to_lt *)
  34 lemma ylt_inv_zero_minus1 (m) (y):
  35       (𝟎) < y - m → yinj_nat m < y.
  36 #m #y @(ynat_split_nat_inf … y) -y //
  37 #n <yminus1_inj_sn >yinj_nat_zero >(nminus_refl m) #Hmm
  38 /4 width=2 by ylt_inv_inj_bi, ylt_inj, nlt_inv_minus_bi_dx/
  39 qed-.
  40
  41 (* Destructions with yminus1 ************************************************)
  42
  43 (*** yminus_pred *)
  44 lemma yminus1_pred_bi (x:ynat) (n):
  45       (𝟎) < x → 𝟎 < n → x - n = ↓x - ↓n.
  46 #x @(ynat_split_nat_inf … x) -x //
  47 #m #n >yinj_nat_zero
  48 #Hm #Hn <yminus1_inj_sn <ypred_inj <yminus1_inj_sn
  49 <nminus_pred_bi /2 width=1 by ylt_inv_inj_bi/
  50 qed-.