(* 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) == case m of 0 => 0 | Suc n => n"
add_def "m+n == nat_rec n (%u v. Suc(v)) m"
diff_def "m-n == nat_rec m (%u v. pred(v)) n"
mult_def "m*n == nat_rec 0 (%u v. n + v) m"
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(0, %z w.z, m) is pred(m). *)