--- a/src/HOL/Number_Theory/Euclidean_Algorithm.thy Sun Feb 28 12:05:52 2016 +0100
+++ b/src/HOL/Number_Theory/Euclidean_Algorithm.thy Sun Feb 28 21:19:58 2016 +0100
@@ -234,6 +234,8 @@
class euclidean_ring = euclidean_semiring + idom
begin
+subclass ring_div ..
+
function euclid_ext_aux :: "'a \<Rightarrow> _" where
"euclid_ext_aux r' r s' s t' t = (
if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r')
@@ -305,6 +307,10 @@
"case euclid_ext x y of (a,b,c) \<Rightarrow> a * x + b * y = c \<and> c = gcd_eucl x y"
unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all
+lemma euclid_ext_gcd_eucl:
+ "(case euclid_ext x y of (a,b,c) \<Rightarrow> c) = gcd_eucl x y"
+ using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold)
+
definition euclid_ext' where
"euclid_ext' x y = (case euclid_ext x y of (a, b, _) \<Rightarrow> (a, b))"