--- a/src/HOL/Rational.thy Fri Nov 20 00:06:04 2009 -0800
+++ b/src/HOL/Rational.thy Fri Nov 20 09:22:14 2009 +0100
@@ -359,17 +359,11 @@
from gcd2 have gcd4: "gcd b' a' = 1"
by (simp add: gcd_commute_int[of a' b'])
have one: "num dvd b'"
- by (rule coprime_dvd_mult_int[OF gcd3], simp add: dvd_def, rule exI[of _ a'], simp add: eq2 algebra_simps)
- have two: "b' dvd num" by (rule coprime_dvd_mult_int[OF gcd4], simp add: dvd_def, rule exI[of _ den], simp add: eq2 algebra_simps)
- from one two
- obtain k k' where k: "num = b' * k" and k': "b' = num * k'"
- unfolding dvd_def by auto
- hence "num = num * (k * k')" by (simp add: algebra_simps)
- with num0 have prod: "k * k' = 1" by auto
- from zero_less_mult_iff[of b' k] b'p num k have kp: "k > 0"
- by auto
- from prod pos_zmult_eq_1_iff[OF kp, of k'] have "k = 1" by auto
- with k have numb': "num = b'" by auto
+ by (metis coprime_dvd_mult_int[OF gcd3] dvd_triv_right eq2)
+ have two: "b' dvd num"
+ by (metis coprime_dvd_mult_int[OF gcd4] dvd_triv_left eq2 zmult_commute)
+ from zdvd_antisym_abs[OF one two] b'p num
+ have numb': "num = b'" by auto
with eq2 b'0 have "a' = den" by auto
with numb' adiv bdiv Pair show ?thesis by simp
qed