(* ID: $Id: Quicksort.thy,v 1.2 2007/10/16 21:12:52 haftmann Exp $ Author: Tobias Nipkow Copyright 1994 TU Muenchen *) header{*Quicksort*} theory Quicksort imports Multiset begin context linorder begin function quicksort :: "'a list => 'a list" where "quicksort [] = []" | "quicksort (x#xs) = quicksort([y\<leftarrow>xs. ~ x≤y]) @ [x] @ quicksort([y\<leftarrow>xs. x≤y])" by pat_completeness auto termination by (relation "measure size") (auto simp: length_filter_le[THEN order_class.le_less_trans]) end context linorder begin lemma quicksort_permutes [simp]: "multiset_of (quicksort xs) = multiset_of xs" by (induct xs rule: quicksort.induct) (auto simp: union_ac) lemma set_quicksort [simp]: "set (quicksort xs) = set xs" 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 end end
lemma quicksort_permutes:
multiset_of (quicksort xs) = multiset_of xs
lemma set_quicksort:
set (quicksort xs) = set xs
lemma sorted_quicksort:
sorted (quicksort xs)