--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Arith.thy	Thu Sep 16 12:21:07 1993 +0200
@@ -0,0 +1,26 @@
+(*  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 +
+arities nat::plus
+        nat::minus
+        nat::times
+consts
+  div,mod  :: "[nat,nat]=>nat"	(infixl 70)
+rules
+  add_def   "m+n == nat_rec(m, n, %u v.Suc(v))"  
+  diff_def  "m-n == nat_rec(n, m, %u v. nat_rec(v, 0, %x y.x))"  
+  mult_def  "m*n == nat_rec(m, 0, %u v. n + v)"  
+  mod_def   "m mod n == wfrec(trancl(pred_nat), m, %j f. if(j<n, j, f(j-n)))"  
+  div_def   "m div n == wfrec(trancl(pred_nat), m, %j f. if(j<n, 0, Suc(f(j-n))))"
+end
+
+(*"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).   *)