src/HOL/Number_Theory/Euclidean_Algorithm.thy

 end
 
 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')
      else let q = r' div r
           in  euclid_ext_aux r (r' mod r) s (s' - q * s) t (t' - q * t))"

 
 lemma euclid_ext_correct':
   "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))"
 
 lemma euclid_ext'_correct':
   "case euclid_ext' x y of (a,b) \<Rightarrow> a * x + b * y = gcd_eucl x y"