header {* Quicksort with function package *}
theory Quicksort
imports "~~/src/HOL/Library/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 sort_quicksort:
"sort = quicksort"
by (rule ext, rule properties_for_sort) (fact quicksort_permutes sorted_quicksort)+
end
end