diff -r c9bba9dba73b -r f91766530e13 src/HOL/GCD.thy
--- a/src/HOL/GCD.thy Tue Sep 20 14:51:58 2016 +0200
+++ b/src/HOL/GCD.thy Sun Sep 18 17:57:55 2016 +0200
@@ -939,6 +939,25 @@
lemma lcm_proj2_iff: "lcm m n = normalize n \ m dvd n"
using lcm_proj1_iff [of n m] by (simp add: ac_simps)
+lemma dvd_productE:
+ assumes "p dvd (a * b)"
+ obtains x y where "p = x * y" "x dvd a" "y dvd b"
+proof (cases "a = 0")
+ case True
+ thus ?thesis by (intro that[of p 1]) simp_all
+next
+ case False
+ define x y where "x = gcd a p" and "y = p div x"
+ have "p = x * y" by (simp add: x_def y_def)
+ moreover have "x dvd a" by (simp add: x_def)
+ moreover from assms have "p dvd gcd (b * a) (b * p)"
+ by (intro gcd_greatest) (simp_all add: mult.commute)
+ hence "p dvd b * gcd a p" by (simp add: gcd_mult_distrib)
+ with False have "y dvd b"
+ by (simp add: x_def y_def div_dvd_iff_mult assms)
+ ultimately show ?thesis by (rule that)
+qed
+
end
class ring_gcd = comm_ring_1 + semiring_gcd