src/HOL/Ring_and_Field.thy

 
 lemma sgn_less [simp]:
   "sgn a < 0 \<longleftrightarrow> a < 0"
   unfolding sgn_if by auto
 
-(* The int instances are proved, these generic ones are tedious to prove here.
-And not very useful, as int seems to be the only instance.
-If needed, they should be proved later, when metis is available.
-lemma dvd_abs[simp]: "(abs m) dvd k \<longleftrightarrow> m dvd k"
-proof-
-  have "\<forall>k.\<exists>ka. - (m * k) = m * ka"
-    by(simp add: mult_minus_right[symmetric] del: mult_minus_right)
-  moreover
-  have "\<forall>k.\<exists>ka. m * k = - (m * ka)"
-    by(auto intro!: minus_minus[symmetric]
-         simp add: mult_minus_right[symmetric] simp del: mult_minus_right)
-  ultimately show ?thesis by (auto simp: abs_if dvd_def)
-qed
-
-lemma dvd_abs2[simp]: "m dvd (abs k) \<longleftrightarrow> m dvd k"
-proof-
-  have "\<forall>k.\<exists>ka. - (m * k) = m * ka"
-    by(simp add: mult_minus_right[symmetric] del: mult_minus_right)
-  moreover
-  have "\<forall>k.\<exists>ka. - (m * ka) = m * k"
-    by(auto intro!: minus_minus
-         simp add: mult_minus_right[symmetric] simp del: mult_minus_right)
-  ultimately show ?thesis
-    by (auto simp add:abs_if dvd_def minus_equation_iff[of k])
-qed
-*)
+lemma abs_dvd_iff [simp]: "(abs m) dvd k \<longleftrightarrow> m dvd k"
+  by (simp add: abs_if)
+
+lemma dvd_abs_iff [simp]: "m dvd (abs k) \<longleftrightarrow> m dvd k"
+  by (simp add: abs_if)
 
 end
 
 class ordered_field = field + ordered_idom