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 *)
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15 include "ground/arith/nat_le_minus.ma".
16 include "ground/arith/nat_lt_pred.ma".
18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
20 (* Rewrites with nminus *****************************************************)
22 (*** minus_pred_pred *)
23 lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m.
25 >(nlt_inv_zero_sn … Hm) in ⊢ (??%?); -Hm
26 >(nlt_inv_zero_sn … Hn) in ⊢ (??%?); -Hn
30 (* Constructions with nminus ************************************************)
32 (*** monotonic_lt_minus_l *)
33 lemma nlt_minus_sn_bi (o) (m) (n): o ≤ m → m < n → m - o < n - o.
35 lapply (nle_minus_sn_bi … o Hmn) -Hmn
36 <(nminus_succ_sn … Hom) //