Added the second half of the W/I correspondence.
(* 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)
(%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"
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). *)