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_le_minus.ma".
16 include "ground/arith/nat_lt_pred.ma".
18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
20 (* Constructions with nminus ************************************************)
22 (*** monotonic_lt_minus_l *)
23 lemma nlt_minus_bi_dx (o) (m) (n): o ≤ m → m < n → m - o < n - o.
25 lapply (nle_minus_bi_dx … o Hmn) -Hmn
26 <(nminus_succ_sn … Hom) //
29 (*** monotonic_lt_minus_r *)
30 lemma nlt_minus_bi_sn (o) (m) (n):
31 m < o -> m < n → o-n < o-m.
33 lapply (nle_minus_bi_sn … o H) -H #H
34 @(nle_nlt_trans … H) -n
35 @nlt_i >(nminus_succ_sn … Ho) //
38 (* Inversions with nminus ***************************************************)
40 lemma nlt_inv_minus_bi_dx (o) (m) (n):
41 m - o < n - o → m < n.
42 #o @(nat_ind_succ … o) -o
43 /3 width=1 by nlt_inv_pred_bi/
46 (* Destructions with nminus *************************************************)
48 (*** minus_pred_pred *)
49 lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m.
51 >(nlt_des_gen … Hm) in ⊢ (??%?); -Hm
52 >(nlt_des_gen … Hn) in ⊢ (??%?); -Hn
56 lemma nlt_des_minus_dx (o) (m) (n): m < n - o → o < n.
57 #o @(nat_ind_succ … o) -o
58 [ #m #n <nminus_zero_dx
59 /2 width=3 by nle_nlt_trans/
60 | #o #IH #m #n <nminus_succ_dx_pred_sn #H
61 /3 width=2 by nlt_inv_pred_dx/