--- a/src/HOL/ex/Qsort.thy Fri Apr 22 15:10:42 2005 +0200
+++ b/src/HOL/ex/Qsort.thy Fri Apr 22 17:32:03 2005 +0200
@@ -2,31 +2,33 @@
ID: $Id$
Author: Tobias Nipkow
Copyright 1994 TU Muenchen
-
-Quicksort
*)
-theory Qsort = Sorting:
+header{*Quicksort*}
-(*Version one: higher-order*)
+theory Qsort
+imports Sorting
+begin
+
+subsection{*Version 1: higher-order*}
+
consts qsort :: "('a \<Rightarrow> 'a => bool) * 'a list \<Rightarrow> 'a list"
recdef qsort "measure (size o snd)"
-"qsort(le, []) = []"
-"qsort(le, x#xs) = qsort(le, [y:xs . ~ le x y]) @ [x] @
- qsort(le, [y:xs . le x y])"
+ "qsort(le, []) = []"
+ "qsort(le, x#xs) = qsort(le, [y:xs . ~ le x y]) @ [x] @
+ qsort(le, [y:xs . le x y])"
(hints recdef_simp: length_filter_le[THEN le_less_trans])
-lemma qsort_permutes[simp]:
- "multiset_of (qsort(le,xs)) = multiset_of xs"
+lemma qsort_permutes [simp]:
+ "multiset_of (qsort(le,xs)) = multiset_of xs"
by (induct le xs rule: qsort.induct) (auto simp: union_ac)
-(*Also provable by induction*)
-lemma set_qsort[simp]: "set (qsort(le,xs)) = set xs";
+lemma set_qsort [simp]: "set (qsort(le,xs)) = set xs";
by(simp add: set_count_greater_0)
lemma sorted_qsort:
- "total(le) ==> transf(le) ==> sorted le (qsort(le,xs))"
+ "total(le) ==> transf(le) ==> sorted le (qsort(le,xs))"
apply (induct le xs rule: qsort.induct)
apply simp
apply simp
@@ -35,28 +37,23 @@
done
-(*Version two: type classes*)
+subsection{*Version 2:type classes*}
consts quickSort :: "('a::linorder) list => 'a list"
recdef quickSort "measure size"
-"quickSort [] = []"
-"quickSort (x#l) = quickSort [y:l. ~ x<=y] @ [x] @ quickSort [y:l. x<=y]"
+ "quickSort [] = []"
+ "quickSort (x#l) = quickSort [y:l. ~ x\<le>y] @ [x] @ quickSort [y:l. x\<le>y]"
(hints recdef_simp: length_filter_le[THEN le_less_trans])
lemma quickSort_permutes[simp]:
"multiset_of (quickSort xs) = multiset_of xs"
by (induct xs rule: quickSort.induct) (auto simp: union_ac)
-(*Also provable by induction*)
lemma set_quickSort[simp]: "set (quickSort xs) = set xs"
by(simp add: set_count_greater_0)
-lemma sorted_quickSort: "sorted (op <=) (quickSort xs)"
-apply (induct xs rule: quickSort.induct)
- apply simp
-apply simp
-apply(blast intro: linorder_linear[THEN disjE] order_trans)
-done
+theorem sorted_quickSort: "sorted (op \<le>) (quickSort xs)"
+by (induct xs rule: quickSort.induct, auto)
end