(* Title: HOL/ex/Primes.thy
ID: $Id$
Author: Christophe Tabacznyj and Lawrence C Paulson
Copyright 1996 University of Cambridge
The Greatest Common Divisor and Euclid's algorithm
The "simpset" clause in the recdef declaration is omitted on purpose:
in order to demonstrate the treatment of definitions that have
unproved termination conditions. Restoring the clause lets
Isabelle prove those conditions.
*)
Primes = Main +
consts
is_gcd :: [nat,nat,nat]=>bool (*gcd as a relation*)
gcd :: "nat*nat=>nat" (*Euclid's algorithm *)
coprime :: [nat,nat]=>bool
prime :: nat set
recdef gcd "measure ((%(m,n).n) ::nat*nat=>nat)"
(* simpset "simpset() addsimps [mod_less_divisor, zero_less_eq]" *)
"gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
defs
is_gcd_def "is_gcd p m n == p dvd m & p dvd n &
(ALL d. d dvd m & d dvd n --> d dvd p)"
coprime_def "coprime m n == gcd(m,n) = 1"
prime_def "prime == {p. 1<p & (ALL m. m dvd p --> m=1 | m=p)}"
end