--- a/src/HOL/GCD.thy Fri Jan 01 17:21:44 2010 +0100
+++ b/src/HOL/GCD.thy Fri Jan 01 19:15:43 2010 +0100
@@ -1689,11 +1689,10 @@
"inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
\<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
apply(auto simp add:inj_on_def)
-apply (metis coprime_dvd_mult_nat dvd.eq_iff gcd_lcm_lattice_nat.inf_sup_absorb
- gcd_semilattice_nat.inf_le2 lcm_proj2_iff_nat nat_mult_1 prod_gcd_lcm_nat)
-apply (metis coprime_dvd_mult_nat gcd_proj1_if_dvd_nat
- gcd_semilattice_nat.inf_commute lcm_dvd1_nat nat_mult_1
- nat_mult_commute prod_gcd_lcm_nat)
+apply (metis gcd_semilattice_nat.inf_commute coprime_dvd_mult_iff_nat
+ dvd.neq_le_trans dvd_triv_left)
+apply (metis gcd_semilattice_nat.inf_commute coprime_dvd_mult_iff_nat
+ dvd.neq_le_trans dvd_triv_right mult_commute)
done
text{* Nitpick: *}