diff --git a/src/HOL/Examples/Gauss_Numbers.thy b/src/HOL/Examples/Gauss_Numbers.thy
--- a/src/HOL/Examples/Gauss_Numbers.thy
+++ b/src/HOL/Examples/Gauss_Numbers.thy
@@ -1,329 +1,340 @@
 (*   Author:      Florian Haftmann, TU Muenchen; based on existing material on complex numbers\<close>
 *)
 
 section \<open>Gauss Numbers: integral gauss numbers\<close>
 
 theory Gauss_Numbers
   imports "HOL-Library.Rounded_Division"
 begin
 
 codatatype gauss = Gauss (Re: int) (Im: int)
 
 lemma gauss_eqI [intro?]:
   \<open>x = y\<close> if \<open>Re x = Re y\<close> \<open>Im x = Im y\<close>
   by (rule gauss.expand) (use that in simp)
 
 lemma gauss_eq_iff:
   \<open>x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y\<close>
   by (auto intro: gauss_eqI)
 
 
 subsection \<open>Basic arithmetic\<close>
 
 instantiation gauss :: comm_ring_1
 begin
 
 primcorec zero_gauss :: \<open>gauss\<close>
   where
     \<open>Re 0 = 0\<close>
   | \<open>Im 0 = 0\<close>
 
 primcorec one_gauss :: \<open>gauss\<close>
   where
     \<open>Re 1 = 1\<close>
   | \<open>Im 1 = 0\<close>
 
 primcorec plus_gauss :: \<open>gauss \<Rightarrow> gauss \<Rightarrow> gauss\<close>
   where
     \<open>Re (x + y) = Re x + Re y\<close>
   | \<open>Im (x + y) = Im x + Im y\<close>
 
 primcorec uminus_gauss :: \<open>gauss \<Rightarrow> gauss\<close>
   where
     \<open>Re (- x) = - Re x\<close>
   | \<open>Im (- x) = - Im x\<close>
 
 primcorec minus_gauss :: \<open>gauss \<Rightarrow> gauss \<Rightarrow> gauss\<close>
   where
     \<open>Re (x - y) = Re x - Re y\<close>
   | \<open>Im (x - y) = Im x - Im y\<close>
 
 primcorec times_gauss :: \<open>gauss \<Rightarrow> gauss \<Rightarrow> gauss\<close>
   where
     \<open>Re (x * y) = Re x * Re y - Im x * Im y\<close>
   | \<open>Im (x * y) = Re x * Im y + Im x * Re y\<close>
 
 instance
   by standard (simp_all add: gauss_eq_iff algebra_simps)
 
 end
 
 lemma of_nat_gauss:
   \<open>of_nat n = Gauss (int n) 0\<close>
   by (induction n) (simp_all add: gauss_eq_iff)
 
 lemma numeral_gauss:
   \<open>numeral n = Gauss (numeral n) 0\<close>
 proof -
   have \<open>numeral n = (of_nat (numeral n) :: gauss)\<close>
     by simp
   also have \<open>\<dots> = Gauss (of_nat (numeral n)) 0\<close>
     by (simp add: of_nat_gauss)
   finally show ?thesis
     by simp
 qed
 
 lemma of_int_gauss:
   \<open>of_int k = Gauss k 0\<close>
   by (simp add: gauss_eq_iff of_int_of_nat of_nat_gauss)
 
 lemma conversion_simps [simp]:
   \<open>Re (numeral m) = numeral m\<close>
   \<open>Im (numeral m) = 0\<close>
   \<open>Re (of_nat n) = int n\<close>
   \<open>Im (of_nat n) = 0\<close>
   \<open>Re (of_int k) = k\<close>
   \<open>Im (of_int k) = 0\<close>
   by (simp_all add: numeral_gauss of_nat_gauss of_int_gauss)
 
 lemma gauss_eq_0:
   \<open>z = 0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0\<close>
   by (simp add: gauss_eq_iff sum_power2_eq_zero_iff)
 
 lemma gauss_neq_0:
   \<open>z \<noteq> 0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0\<close>
   by (simp add: gauss_eq_0 sum_power2_ge_zero less_le)
 
 lemma Re_sum [simp]:
   \<open>Re (sum f s) = (\<Sum>x\<in>s. Re (f x))\<close>
   by (induct s rule: infinite_finite_induct) auto
 
 lemma Im_sum [simp]:
   \<open>Im (sum f s) = (\<Sum>x\<in>s. Im (f x))\<close>
   by (induct s rule: infinite_finite_induct) auto
 
 instance gauss :: idom
 proof
   fix x y :: gauss
   assume \<open>x \<noteq> 0\<close> \<open>y \<noteq> 0\<close>
   then show \<open>x * y \<noteq> 0\<close>
     by (simp_all add: gauss_eq_iff)
       (smt (verit, best) mult_eq_0_iff mult_neg_neg mult_neg_pos mult_pos_neg mult_pos_pos)
 qed
 
 
 
 subsection \<open>The Gauss Number $i$\<close>
 
 primcorec imaginary_unit :: gauss  (\<open>\<i>\<close>)
   where
     \<open>Re \<i> = 0\<close>
   | \<open>Im \<i> = 1\<close>
 
 lemma Gauss_eq:
   \<open>Gauss a b = of_int a + \<i> * of_int b\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_eq:
   \<open>a = of_int (Re a) + \<i> * of_int (Im a)\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_i_not_zero [simp]:
   \<open>\<i> \<noteq> 0\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_i_not_one [simp]:
   \<open>\<i> \<noteq> 1\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_i_not_numeral [simp]:
   \<open>\<i> \<noteq> numeral n\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_i_not_neg_numeral [simp]:
   \<open>\<i> \<noteq> - numeral n\<close>
   by (simp add: gauss_eq_iff)
 
 lemma i_mult_i_eq [simp]:
   \<open>\<i> * \<i> = - 1\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_i_mult_minus [simp]:
   \<open>\<i> * (\<i> * x) = - x\<close>
   by (simp flip: mult.assoc)
 
 lemma i_squared [simp]:
   \<open>\<i>\<^sup>2 = - 1\<close>
   by (simp add: power2_eq_square)
 
 lemma i_even_power [simp]:
   \<open>\<i> ^ (n * 2) = (- 1) ^ n\<close>
   unfolding mult.commute [of n] power_mult by simp
 
 lemma Re_i_times [simp]:
   \<open>Re (\<i> * z) = - Im z\<close>
   by simp
 
 lemma Im_i_times [simp]:
   \<open>Im (\<i> * z) = Re z\<close>
   by simp
 
 lemma i_times_eq_iff:
   \<open>\<i> * w = z \<longleftrightarrow> w = - (\<i> * z)\<close>
   by auto
 
 lemma is_unit_i [simp]:
   \<open>\<i> dvd 1\<close>
   by (rule dvdI [of _ _ \<open>- \<i>\<close>]) simp
 
 lemma gauss_numeral [code_post]:
   \<open>Gauss 0 0 = 0\<close>
   \<open>Gauss 1 0 = 1\<close>
   \<open>Gauss (- 1) 0 = - 1\<close>
   \<open>Gauss (numeral n) 0 = numeral n\<close>
   \<open>Gauss (- numeral n) 0 = - numeral n\<close>
   \<open>Gauss 0 1 = \<i>\<close>
   \<open>Gauss 0 (- 1) = - \<i>\<close>
   \<open>Gauss 0 (numeral n) = numeral n * \<i>\<close>
   \<open>Gauss 0 (- numeral n) = - numeral n * \<i>\<close>
   \<open>Gauss 1 1 = 1 + \<i>\<close>
   \<open>Gauss (- 1) 1 = - 1 + \<i>\<close>
   \<open>Gauss (numeral n) 1 = numeral n + \<i>\<close>
   \<open>Gauss (- numeral n) 1 = - numeral n + \<i>\<close>
   \<open>Gauss 1 (- 1) = 1 - \<i>\<close>
   \<open>Gauss 1 (numeral n) = 1 + numeral n * \<i>\<close>
   \<open>Gauss 1 (- numeral n) = 1 - numeral n * \<i>\<close>
   \<open>Gauss (- 1) (- 1) = - 1 - \<i>\<close>
   \<open>Gauss (numeral n) (- 1) = numeral n - \<i>\<close>
   \<open>Gauss (- numeral n) (- 1) = - numeral n - \<i>\<close>
   \<open>Gauss (- 1) (numeral n) = - 1 + numeral n * \<i>\<close>
   \<open>Gauss (- 1) (- numeral n) = - 1 - numeral n * \<i>\<close>
   \<open>Gauss (numeral m) (numeral n) = numeral m + numeral n * \<i>\<close>
   \<open>Gauss (- numeral m) (numeral n) = - numeral m + numeral n * \<i>\<close>
   \<open>Gauss (numeral m) (- numeral n) = numeral m - numeral n * \<i>\<close>
   \<open>Gauss (- numeral m) (- numeral n) = - numeral m - numeral n * \<i>\<close>
   by (simp_all add: gauss_eq_iff)
 
 
 subsection \<open>Gauss Conjugation\<close>
 
 primcorec cnj :: \<open>gauss \<Rightarrow> gauss\<close>
   where
     \<open>Re (cnj z) = Re z\<close>
   | \<open>Im (cnj z) = - Im z\<close>
 
 lemma gauss_cnj_cancel_iff [simp]:
   \<open>cnj x = cnj y \<longleftrightarrow> x = y\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_cnj_cnj [simp]:
   \<open>cnj (cnj z) = z\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_cnj_zero [simp]:
   \<open>cnj 0 = 0\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_cnj_zero_iff [iff]:
   \<open>cnj z = 0 \<longleftrightarrow> z = 0\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_cnj_one_iff [simp]:
   \<open>cnj z = 1 \<longleftrightarrow> z = 1\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_cnj_add [simp]:
   \<open>cnj (x + y) = cnj x + cnj y\<close>
   by (simp add: gauss_eq_iff)
 
 lemma cnj_sum [simp]:
   \<open>cnj (sum f s) = (\<Sum>x\<in>s. cnj (f x))\<close>
   by (induct s rule: infinite_finite_induct) auto
 
 lemma gauss_cnj_diff [simp]:
   \<open>cnj (x - y) = cnj x - cnj y\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_cnj_minus [simp]:
   \<open>cnj (- x) = - cnj x\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_cnj_one [simp]:
   \<open>cnj 1 = 1\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_cnj_mult [simp]:
   \<open>cnj (x * y) = cnj x * cnj y\<close>
   by (simp add: gauss_eq_iff)
 
 lemma cnj_prod [simp]:
   \<open>cnj (prod f s) = (\<Prod>x\<in>s. cnj (f x))\<close>
   by (induct s rule: infinite_finite_induct) auto
 
 lemma gauss_cnj_power [simp]:
   \<open>cnj (x ^ n) = cnj x ^ n\<close>
   by (induct n) simp_all
 
 lemma gauss_cnj_numeral [simp]:
   \<open>cnj (numeral w) = numeral w\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_cnj_of_nat [simp]:
   \<open>cnj (of_nat n) = of_nat n\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_cnj_of_int [simp]:
   \<open>cnj (of_int z) = of_int z\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_cnj_i [simp]:
   \<open>cnj \<i> = - \<i>\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_add_cnj:
   \<open>z + cnj z = of_int (2 * Re z)\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_diff_cnj:
   \<open>z - cnj z = of_int (2 * Im z) * \<i>\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_mult_cnj:
   \<open>z * cnj z = of_int ((Re z)\<^sup>2 + (Im z)\<^sup>2)\<close>
   by (simp add: gauss_eq_iff power2_eq_square)
 
 lemma cnj_add_mult_eq_Re:
   \<open>z * cnj w + cnj z * w = of_int (2 * Re (z * cnj w))\<close>
   by (simp add: gauss_eq_iff)
 
 lemma gauss_In_mult_cnj_zero [simp]:
   \<open>Im (z * cnj z) = 0\<close>
   by simp
 
 
 subsection \<open>Algebraic division\<close>
 
 instantiation gauss :: idom_modulo
 begin
 
 primcorec divide_gauss :: \<open>gauss \<Rightarrow> gauss \<Rightarrow> gauss\<close>
   where
     \<open>Re (x div y) = (Re x * Re y + Im x * Im y) rdiv ((Re y)\<^sup>2 + (Im y)\<^sup>2)\<close>
   | \<open>Im (x div y) = (Im x * Re y - Re x * Im y) rdiv ((Re y)\<^sup>2 + (Im y)\<^sup>2)\<close>
 
-definition modulo_gauss :: \<open>gauss \<Rightarrow> gauss \<Rightarrow> gauss\<close>
+primcorec modulo_gauss :: \<open>gauss \<Rightarrow> gauss \<Rightarrow> gauss\<close>
   where
-    \<open>x mod y = x - x div y * y\<close> for x y :: gauss
+    \<open>Re (x mod y) = Re x -
+      ((Re x * Re y + Im x * Im y) rdiv ((Re y)\<^sup>2 + (Im y)\<^sup>2) * Re y -
+       (Im x * Re y - Re x * Im y) rdiv ((Re y)\<^sup>2 + (Im y)\<^sup>2) * Im y)\<close>
+  | \<open>Im (x mod y) = Im x -
+      ((Re x * Re y + Im x * Im y) rdiv ((Re y)\<^sup>2 + (Im y)\<^sup>2) * Im y +
+       (Im x * Re y - Re x * Im y) rdiv ((Re y)\<^sup>2 + (Im y)\<^sup>2) * Re y)\<close>
 
-instance
-  apply standard
-  apply (simp_all add: modulo_gauss_def)
-  apply (auto simp add: gauss_eq_iff algebra_simps power2_eq_square)
-           apply (simp_all only: flip: mult.assoc distrib_right)
-       apply (simp_all only: mult.assoc [of \<open>Im k\<close> \<open>Re l\<close> \<open>Re r\<close> for k l r])
-     apply (simp_all add: sum_squares_eq_zero_iff mult.commute nonzero_mult_rdiv_cancel_right flip: distrib_left)
-  done
+instance proof
+  fix x y :: gauss
+  show \<open>x div 0 = 0\<close>
+    by (simp add: gauss_eq_iff)
+  show \<open>x * y div y = x\<close> if \<open>y \<noteq> 0\<close>
+    using that
+    apply (auto simp add: gauss_eq_iff algebra_simps power2_eq_square)
+       apply (simp_all only: flip: mult.assoc distrib_right)
+    apply (simp_all add: mult.assoc [of _ \<open>Re y\<close> \<open>Re y\<close>] mult.assoc [of _ \<open>Im y\<close> \<open>Im y\<close>] mult.commute [of _ \<open>Re x\<close>] flip: distrib_left)
+       apply (simp_all add: sum_squares_eq_zero_iff mult.commute nonzero_mult_rdiv_cancel_right flip: distrib_left)
+    done
+  show \<open>x div y * y + x mod y = x\<close>
+    by (simp add: gauss_eq_iff)
+qed
 
 end
 
 end