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_pred_succ.ma".
16 include "ground/arith/nat_le.ma".
18 (* ORDER FOR NON-NEGATIVE INTEGERS ******************************************)
20 (* Constructions with npred *************************************************)
23 lemma nle_pred_sn_refl (m): ↓m ≤ m.
27 (*** monotonic_pred *)
28 lemma nle_pred_bi (m) (n): m ≤ n → ↓m ≤ ↓n.
30 /2 width=3 by nle_trans/
33 lemma nle_pred_sn (m) (n): m ≤ ↑n → ↓m ≤ n.
35 /2 width=1 by nle_pred_bi/
38 (* Destructions with npred **************************************************)
40 lemma nle_inv_pred_sn (m) (n): ↓m ≤ n → m ≤ ↑n.
42 /2 width=1 by nle_succ_bi/