src/HOL/ex/MergeSort.thy
 author huffman Fri Aug 19 14:17:28 2011 -0700 (2011-08-19) changeset 44311 42c5cbf68052 parent 41413 64cd30d6b0b8 child 58889 5b7a9633cfa8 permissions -rw-r--r--
new isCont theorems;
simplify some proofs.
1 (*  Title:      HOL/ex/MergeSort.thy
2     Author:     Tobias Nipkow
4 *)
8 theory MergeSort
9 imports "~~/src/HOL/Library/Multiset"
10 begin
12 context linorder
13 begin
15 fun merge :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
16 where
17   "merge (x#xs) (y#ys) =
18          (if x \<le> y then x # merge xs (y#ys) else y # merge (x#xs) ys)"
19 | "merge xs [] = xs"
20 | "merge [] ys = ys"
22 lemma multiset_of_merge [simp]:
23   "multiset_of (merge xs ys) = multiset_of xs + multiset_of ys"
24   by (induct xs ys rule: merge.induct) (simp_all add: ac_simps)
26 lemma set_merge [simp]:
27   "set (merge xs ys) = set xs \<union> set ys"
28   by (induct xs ys rule: merge.induct) auto
30 lemma sorted_merge [simp]:
31   "sorted (merge xs ys) \<longleftrightarrow> sorted xs \<and> sorted ys"
32   by (induct xs ys rule: merge.induct) (auto simp add: ball_Un not_le less_le sorted_Cons)
34 fun msort :: "'a list \<Rightarrow> 'a list"
35 where
36   "msort [] = []"
37 | "msort [x] = [x]"
38 | "msort xs = merge (msort (take (size xs div 2) xs))
39                     (msort (drop (size xs div 2) xs))"
41 lemma sorted_msort:
42   "sorted (msort xs)"
43   by (induct xs rule: msort.induct) simp_all
45 lemma multiset_of_msort:
46   "multiset_of (msort xs) = multiset_of xs"
47   by (induct xs rule: msort.induct)
48     (simp_all, metis append_take_drop_id drop_Suc_Cons multiset_of.simps(2) multiset_of_append take_Suc_Cons)
50 theorem msort_sort:
51   "sort = msort"
52   by (rule ext, rule properties_for_sort) (fact multiset_of_msort sorted_msort)+
54 end
56 end