src/HOL/Arith.thy
 author paulson Fri, 14 Jun 1996 12:22:59 +0200 changeset 1796 c42db9ab8728 parent 1475 7f5a4cd08209 child 1824 44254696843a permissions -rw-r--r--
Tidied spacing
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(*  Title:      HOL/Arith.thy
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

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).   *)

```