author paulson
Fri, 14 Jun 1996 12:22:59 +0200
changeset 1796 c42db9ab8728
parent 1475 7f5a4cd08209
child 1824 44254696843a
permissions -rw-r--r--
Tidied spacing

(*  Title:      HOL/Arith.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge

Arithmetic operators and their definitions

Arith = Nat +

  nat :: {plus, minus, times}

  pred      :: nat => nat
  div, mod  :: [nat, nat] => nat  (infixl 70)

  pred_def  "pred(m) == nat_rec m 0 (%n r.n)"
  add_def   "m+n == nat_rec m n (%u v. Suc(v))"
  diff_def  "m-n == nat_rec n m (%u v. pred(v))"
  mult_def  "m*n == nat_rec m 0 (%u v. n + v)"

  mod_def   "m mod n == wfrec (trancl pred_nat)
                          (%f j. if j<n then j else f (j-n)) m"
  div_def   "m div n == wfrec (trancl pred_nat) 
                          (%f j. if j<n then 0 else Suc (f (j-n))) m"

(*"Difference" is subtraction of natural numbers.
  There are no negative numbers; we have
     m - n = 0  iff  m<=n   and     m - n = Suc(k) iff m)n.
  Also, nat_rec(m, 0, %z w.z) is pred(m).   *)