--- a/src/HOL/Ring_and_Field.thy Mon Feb 16 19:35:52 2009 -0800
+++ b/src/HOL/Ring_and_Field.thy Tue Feb 17 06:59:33 2009 -0800
@@ -1100,32 +1100,11 @@
"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