1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "ground/arith/ynat_plus.ma".
16 include "ground/arith/ynat_lminus.ma".
17 include "ground/arith/ynat_lt_pred_succ.ma".
19 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
21 (* Constructions with ylminus and yplus *************************************)
23 (*** ylt_plus1_to_minus_inj2 *)
24 lemma ylt_plus_sn_dx_lminus_dx (n) (x) (z):
25 x + yinj_nat n < z → x < z - n.
26 #n @(nat_ind_succ … n) -n //
27 #n #IH #x #z >ysucc_inj <yplus_succ_shift
28 /3 width=1 by ylt_des_succ_sn/
31 (*** ylt_plus1_to_minus_inj1 *)
32 lemma ylt_plus_sn_sn_lminus_dx (n) (x) (z):
33 yinj_nat n + x < z → x < z - n.
34 /2 width=1 by ylt_plus_sn_dx_lminus_dx/ qed.