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1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
5 (*      ||T||                                                             *)
6 (*      ||I||       Developers:                                           *)
7 (*      ||T||         The HELM team.                                      *)
8 (*      ||A||         http://helm.cs.unibo.it                             *)
9 (*      \   /                                                             *)
10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU General Public License Version 2                  *)
12 (*                                                                        *)
13 (**************************************************************************)
14
15 include "nat/order.ma".
16
17 nlet rec minus (n:nat) (m:nat) on m : nat ≝
18  match m with
19   [ O ⇒ n
20   | S m' ⇒
21      match n with
22       [ O ⇒ O
23       | S n' ⇒ minus n' m']].
24
25 interpretation "natural minus" 'minus x y = (minus x y).
26
27 naxiom le_minus: ∀n,m,p. le n m → le (minus n p) m.
28 naxiom lt_minus: ∀n,m,p. lt n m → lt (minus n p) m.