src/HOL/ex/Quicksort.thy

+(*  Author:     Tobias Nipkow
+    Copyright   1994 TU Muenchen
+*)
+
+header {* Quicksort with function package *}
+
+theory Quicksort
+imports Main Multiset
+begin
+
+context linorder
+begin
+
+fun quicksort :: "'a list \<Rightarrow> 'a list" where
+  "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) (simp_all add: ac_simps)
+
+lemma set_quicksort [simp]: "set (quicksort xs) = set xs"
+  by (simp add: set_count_greater_0)
+
+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 sort_quicksort:
+  "sort = quicksort"
+  by (rule ext, rule properties_for_sort) (fact quicksort_permutes sorted_quicksort)+
+
+end
+
+end