# HG changeset patch
# User haftmann
# Date 1288713580 -3600
# Node ID ad053b4e2b6de399887077b15c59a786ffeb54b0
# Parent  e4461b9854a5a79da7d36b348a9b86c23b39c177
tuned proof

diff -r e4461b9854a5 -r ad053b4e2b6d src/HOL/Library/Multiset.thy
--- a/src/HOL/Library/Multiset.thy	Tue Nov 02 16:48:19 2010 +0100
+++ b/src/HOL/Library/Multiset.thy	Tue Nov 02 16:59:40 2010 +0100
@@ -901,28 +901,28 @@
 next
   fix l
   assume "l \<in> set ?rhs"
-  let ?pivot = "f (xs ! (length xs div 2))"
   have *: "\<And>x P. P (f x) \<and> f l = f x \<longleftrightarrow> P (f l) \<and> f l = f x" by auto
   have **: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
   with ** have [simp]: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
+  let ?pivot = "f (xs ! (length xs div 2))"
   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
   proof (cases "f l" ?pivot rule: linorder_cases)
-    case less then show ?thesis
+    case less then moreover have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
+    ultimately show ?thesis
       apply (auto simp add: filter_sort [symmetric])
       apply (subst *) apply simp
-      apply (frule less_imp_neq) apply simp
-      apply (subst *) apply (frule less_not_sym) apply simp
+      apply (subst *) apply simp
       done
   next
     case equal then show ?thesis
       by (auto simp add: ** less_le)
   next
-    case greater then show ?thesis
+    case greater then moreover have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
+    ultimately show ?thesis
       apply (auto simp add: filter_sort [symmetric])
-      apply (subst *) apply (frule less_not_sym) apply simp
-      apply (frule less_imp_neq) apply simp
+      apply (subst *) apply simp
       apply (subst *) apply simp
       done
   qed