(* Title: HOL/ex/Primes.thy
ID: $Id$
Author: Christophe Tabacznyj and Lawrence C Paulson
Copyright 1996 University of Cambridge
The "divides" relation, the Greatest Common Divisor and Euclid's algorithm
*)
Primes = Arith + WF_Rel +
consts
dvd :: [nat,nat]=>bool (infixl 70)
is_gcd :: [nat,nat,nat]=>bool (*gcd as a relation*)
gcd :: "nat*nat=>nat" (*Euclid's algorithm *)
coprime :: [nat,nat]=>bool
prime :: nat=>bool
recdef gcd "measure ((%(x,y).y) ::nat*nat=>nat)"
"gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
defs
dvd_def "m dvd n == EX k. n = m*k"
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(n) == (1<n) & (ALL m. 1<m & m<n --> ~(m dvd n))"
end