Theory Quicksort

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theory Quicksort
imports Multiset

(*  Author:     Tobias Nipkow
Copyright 1994 TU Muenchen
*)


header {* Quicksort *}

theory Quicksort
imports Main Multiset
begin


context linorder
begin


fun quicksort :: "'a list => 'a list" where
"quicksort [] = []"
| "quicksort (x#xs) = quicksort [y\<leftarrow>xs. ¬ x≤y] @ [x] @ quicksort [y\<leftarrow>xs. x≤y]"


lemma [code]:
"quicksort [] = []"
"quicksort (x#xs) = quicksort [y\<leftarrow>xs. y<x] @ [x] @ quicksort [y\<leftarrow>xs. x≤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 quicksort_sort [code_unfold]:
"sort = quicksort"

by (rule ext, rule properties_for_sort) (fact quicksort_permutes sorted_quicksort)+

end

end