# Theory Quicksort

theory Quicksort
imports Multiset
```(*  Author:     Tobias Nipkow
*)

section ‹Quicksort with function package›

theory Quicksort
imports "HOL-Library.Multiset"
begin

context linorder
begin

fun quicksort :: "'a list ⇒ 'a list" where
"quicksort []     = []"
| "quicksort (x#xs) = quicksort [y←xs. ¬ x≤y] @ [x] @ quicksort [y←xs. x≤y]"

lemma [code]:
"quicksort []     = []"
"quicksort (x#xs) = quicksort [y←xs. y<x] @ [x] @ quicksort [y←xs. x≤y]"

lemma quicksort_permutes [simp]:
"mset (quicksort xs) = mset xs"
by (induct xs rule: quicksort.induct) (simp_all add: ac_simps)

lemma set_quicksort [simp]: "set (quicksort xs) = set xs"
proof -
have "set_mset (mset (quicksort xs)) = set_mset (mset xs)"
by simp
then show ?thesis by (simp only: set_mset_mset)
qed

lemma sorted_quicksort: "sorted (quicksort xs)"
by (induct xs rule: quicksort.induct) (auto simp add: 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
```