--- a/src/HOL/SupInf.thy Thu Mar 18 13:56:34 2010 +0100
+++ b/src/HOL/SupInf.thy Thu Mar 18 13:56:34 2010 +0100
@@ -355,13 +355,13 @@
lemma Inf_greater:
fixes z :: real
- shows "X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x \<in> X. x < z"
+ shows "X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x \<in> X. x < z"
by (metis Inf_real_iff mem_def not_leE)
lemma Inf_close:
fixes e :: real
shows "X \<noteq> {} \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>x \<in> X. x < Inf X + e"
- by (metis add_strict_increasing comm_monoid_add.mult_commute Inf_greater linorder_not_le pos_add_strict)
+ by (metis add_strict_increasing add_commute Inf_greater linorder_not_le pos_add_strict)
lemma Inf_finite_Min:
fixes S :: "real set"
@@ -417,7 +417,7 @@
also have "... \<le> e"
apply (rule Sup_asclose)
apply (auto simp add: S)
- apply (metis abs_minus_add_cancel b comm_monoid_add.mult_commute real_diff_def)
+ apply (metis abs_minus_add_cancel b add_commute real_diff_def)
done
finally have "\<bar>- Sup (uminus ` S) - l\<bar> \<le> e" .
thus ?thesis