(* 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 +
instance
nat :: {plus, minus, times}
consts
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)))"
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). *)