helm/software/matita/nlibrary/nat/minus.ma

   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 naxiom le_minus: ∀n,m,p. le n m → le (minus n p) m.
  26 naxiom lt_minus: ∀n,m,p. lt n m → lt (minus n p) m.