--- a/src/HOL/Library/Quicksort.thy Sat May 22 10:12:49 2010 +0200
+++ b/src/HOL/Library/Quicksort.thy Sat May 22 10:12:50 2010 +0200
@@ -2,7 +2,7 @@
Copyright 1994 TU Muenchen
*)
-header{*Quicksort*}
+header {* Quicksort *}
theory Quicksort
imports Main Multiset
@@ -12,22 +12,27 @@
begin
fun quicksort :: "'a list \<Rightarrow> 'a list" where
-"quicksort [] = []" |
-"quicksort (x#xs) = quicksort([y\<leftarrow>xs. ~ x\<le>y]) @ [x] @ quicksort([y\<leftarrow>xs. x\<le>y])"
+ "quicksort [] = []"
+| "quicksort (x#xs) = quicksort [y\<leftarrow>xs. \<not> x\<le>y] @ [x] @ quicksort [y\<leftarrow>xs. x\<le>y]"
+
+lemma [code]:
+ "quicksort [] = []"
+ "quicksort (x#xs) = quicksort [y\<leftarrow>xs. y<x] @ [x] @ quicksort [y\<leftarrow>xs. x\<le>y]"
+ by (simp_all add: not_le)
lemma quicksort_permutes [simp]:
"multiset_of (quicksort xs) = multiset_of xs"
-by (induct xs rule: quicksort.induct) (auto simp: union_ac)
+ by (induct xs rule: quicksort.induct) (simp_all add: ac_simps)
lemma set_quicksort [simp]: "set (quicksort xs) = set xs"
-by(simp add: set_count_greater_0)
+ by (simp add: set_count_greater_0)
-lemma sorted_quicksort: "sorted(quicksort xs)"
-apply (induct xs rule: quicksort.induct)
- apply simp
-apply (simp add:sorted_Cons sorted_append not_le less_imp_le)
-apply (metis leD le_cases le_less_trans)
-done
+lemma sorted_quicksort: "sorted (quicksort xs)"
+ by (induct xs rule: quicksort.induct) (auto simp add: sorted_Cons sorted_append not_le less_imp_le)
+
+theorem quicksort_sort [code_unfold]:
+ "sort = quicksort"
+ by (rule ext, rule properties_for_sort) (fact quicksort_permutes sorted_quicksort)+
end