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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   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/ynat_succ.ma".
  16 include "ground/arith/ynat_lt_pred.ma".
  17
  18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
  19
  20 (* Constructions with ypred and ysucc ***************************************)
  21
  22 (*** ylt_O1 *)
  23 lemma ylt_zero_sn (y): y = ↑↓y → 𝟎 < y.
  24 #y @(ynat_split_nat_inf … y) -y
  25 /4 width=1 by ylt_inj, eq_inv_yinj_nat_bi, nlt_zero_sn/
  26 qed.
  27
  28 (* Destructions with ypred and ysucc ****************************************)
  29
  30 (*** ylt_inv_O1 *)
  31 lemma ylt_des_gen_dx (x) (y): x < y → y = ↑↓y.
  32 #x #y * //
  33 #m #n #H
  34 lapply (nlt_des_gen … H) -H //
  35 qed-.
  36
  37 lemma ylt_des_succ_sn (x) (y):
  38       ↑x < y → x < ↓y.
  39 #x #y @(insert_eq_1 … (↑x))
  40 #x0 * -x0 -y
  41 [ #m0 #n #Hn #H
  42   elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct
  43   elim (nlt_inv_succ_sn … Hn) -Hn #Hm #_
  44   /2 width=1 by ylt_inj/
  45 | #m0 #H
  46   elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct //
  47 ]
  48 qed-.
  49
  50 (* Inversions with ypred and ysucc ******************************************)
  51
  52 (*** ylt_inv_succ1 *)
  53 lemma ylt_inv_succ_sn (x) (y):
  54       ↑x < y → ∧∧ x < ↓y & y = ↑↓y.
  55 /3 width=2 by ylt_des_succ_sn, ylt_des_gen_dx, conj/ qed-.