1.1 --- a/src/HOL/Rational.thy	Fri Nov 20 00:06:04 2009 -0800
     1.2 +++ b/src/HOL/Rational.thy	Fri Nov 20 09:22:14 2009 +0100
     1.3 @@ -359,17 +359,11 @@
     1.4              from gcd2 have gcd4: "gcd b' a' = 1"
     1.5                by (simp add: gcd_commute_int[of a' b'])
     1.6              have one: "num dvd b'"
     1.7 -              by (rule coprime_dvd_mult_int[OF gcd3], simp add: dvd_def, rule exI[of _ a'], simp add: eq2 algebra_simps)
     1.8 -            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)
     1.9 -            from one two
    1.10 -            obtain k k' where k: "num = b' * k" and k': "b' = num * k'"
    1.11 -              unfolding dvd_def by auto
    1.12 -            hence "num = num * (k * k')" by (simp add: algebra_simps)
    1.13 -            with num0 have prod: "k * k' = 1" by auto
    1.14 -            from zero_less_mult_iff[of b' k] b'p num k have kp: "k > 0"
    1.15 -              by auto
    1.16 -            from prod pos_zmult_eq_1_iff[OF kp, of k'] have "k = 1" by auto
    1.17 -            with k have numb': "num = b'" by auto
    1.18 +              by (metis coprime_dvd_mult_int[OF gcd3] dvd_triv_right eq2)
    1.19 +            have two: "b' dvd num"
    1.20 +              by (metis coprime_dvd_mult_int[OF gcd4] dvd_triv_left eq2 zmult_commute)
    1.21 +            from zdvd_antisym_abs[OF one two] b'p num
    1.22 +            have numb': "num = b'" by auto
    1.23              with eq2 b'0 have "a' = den" by auto
    1.24              with numb' adiv bdiv Pair show ?thesis by simp
    1.25            qed