--- a/Arith.thy Fri Sep 16 15:48:20 1994 +0200
+++ b/Arith.thy Wed Sep 21 15:40:41 1994 +0200
@@ -1,24 +1,26 @@
-(* Title: HOL/arith.thy
+(* Title: HOL/Arith.thy
ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ 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
+
+instance
+ nat :: {plus, minus, times}
+
consts
- pred :: "nat => nat"
- div,mod :: "[nat,nat]=>nat" (infixl 70)
-rules
+ pred :: "nat => nat"
+ div, mod :: "[nat, nat] => nat" (infixl 70)
+
+defs
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), m, %j f. if(j<n, j, f(j-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), 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
@@ -26,3 +28,4 @@
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). *)
+