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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  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_pred_succ.ma".
  16 include "ground/arith/ynat_plus.ma".
  17 include "ground/arith/ynat_lt_pred_succ.ma".
  18
  19 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
  20
  21 (* Inversions with yplus and ypred ******************************************)
  22
  23 (*** yplus_inv_succ_lt_dx *)
  24 lemma eq_inv_succ_yplus_lt_dx (z) (x) (y):  𝟎 < y → ↑z = x + y → z = x + ↓y.
  25 #z #x #y #Hy >(ylt_des_gen_dx … Hy) -Hy
  26 <yplus_succ_dx <ypred_succ
  27 /2 width=1 by eq_inv_ysucc_bi/
  28 qed-.
  29
  30 (*** yplus_inv_succ_lt_sn *)
  31 lemma eq_inv_succ_yplus_lt_sn (z) (x) (y): 𝟎 < x → ↑z = x + y → z = ↓x + y.
  32 /2 width=1 by eq_inv_succ_yplus_lt_dx/ qed-.
  33
  34 (* Destructions with yplus and ypred ****************************************)
  35
  36 (*** yplus_pred1 *)
  37 lemma yplus_pred_sn (x) (y): 𝟎 < x → ↓(x+y) = ↓x + y.
  38 #x #y #Hx >(ylt_des_gen_dx … Hx) -Hx
  39 <yplus_succ_sn <ypred_succ <ypred_succ //
  40 qed-.
  41
  42 (*** yplus_pred2 *)
  43 lemma yplus_pred_dx (x) (y): 𝟎 < y → x + ↓y = ↓(x+y).
  44 /2 width=1 by yplus_pred_sn/ qed-.
  45