# HG changeset patch
# User haftmann
# Date 1288781429 -3600
# Node ID 58af2b8327b77850b7df9a5ea57ed75836335257
# Parent  abc52faa7761eabb46a4c141b7e04dd55608222a
tuned proof

diff -r abc52faa7761 -r 58af2b8327b7 src/HOL/Library/Multiset.thy
--- a/src/HOL/Library/Multiset.thy	Wed Nov 03 10:44:53 2010 +0100
+++ b/src/HOL/Library/Multiset.thy	Wed Nov 03 11:50:29 2010 +0100
@@ -901,30 +901,27 @@
 next
   fix l
   assume "l \<in> set ?rhs"
-  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
+  let ?pivot = "f (xs ! (length xs div 2))"
+  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))"
+  with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
+  have "\<And>x P. P (f x) ?pivot \<and> f l = f x \<longleftrightarrow> P (f l) ?pivot \<and> f l = f x" by auto
+  then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
+    [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
+  note *** = this [of "op <"] this [of "op >"] this [of "op ="]
   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 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 (subst *) apply simp
-      done
+      by (simp add: filter_sort [symmetric] ** ***)
   next
     case equal then show ?thesis
-      by (auto simp add: ** less_le)
+      by (simp add: * less_le)
   next
     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 simp
-      apply (subst *) apply simp
-      done
+      by (simp add: filter_sort [symmetric] ** ***)
   qed
 qed