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/nat_lt_minus.ma".
16 include "ground/arith/ynat_lminus.ma".
17 include "ground/arith/ynat_lt.ma".
19 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
21 (* Constructions with ylminus ***********************************************)
24 lemma ylt_zero_lminus (m) (y):
25 yinj_nat m < y → 𝟎 < y - m.
26 #m #y @(ynat_split_nat_inf … y) -y //
27 #n #Hmn <ylminus_inj_sn >yinj_nat_zero >(nminus_refl m)
28 /4 width=1 by ylt_inj, ylt_inv_inj_bi, nlt_minus_bi_dx/
31 (* Inversions with ylminus **************************************************)
34 lemma ylt_inv_zero_lminus (m) (y):
35 (𝟎) < y - m → yinj_nat m < y.
36 #m #y @(ynat_split_nat_inf … y) -y //
37 #n <ylminus_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/
41 (* Destructions with ylminus ************************************************)
44 lemma ylminus_pred_bi (x:ynat) (n):
45 (𝟎) < x → 𝟎 < n → x - n = ↓x - ↓n.
46 #x @(ynat_split_nat_inf … x) -x //
48 #Hm #Hn <ylminus_inj_sn <ypred_inj <ylminus_inj_sn
49 <nminus_pred_bi /2 width=1 by ylt_inv_inj_bi/