author haftmann Wed, 03 Nov 2010 11:50:29 +0100 changeset 40346 58af2b8327b7 parent 40320 abc52faa7761 child 40347 429bf4388b2f
tuned proof
```--- 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
```