author haftmann Sat, 22 May 2010 10:13:02 +0200 changeset 37076 4d57f872dc2c parent 37075 a680ce27aa56 child 37077 3b247fa77c68
modernized sorting algorithms; quicksort implements sort
 src/HOL/ex/MergeSort.thy file | annotate | diff | comparison | revisions
```--- a/src/HOL/ex/MergeSort.thy	Sat May 22 10:12:50 2010 +0200
+++ b/src/HOL/ex/MergeSort.thy	Sat May 22 10:13:02 2010 +0200
@@ -6,7 +6,7 @@

theory MergeSort
-imports Sorting
+imports Multiset
begin

context linorder
@@ -19,23 +19,17 @@
| "merge xs [] = xs"
| "merge [] ys = ys"

-lemma multiset_of_merge[simp]:
-     "multiset_of (merge xs ys) = multiset_of xs + multiset_of ys"
-apply(induct xs ys rule: merge.induct)
-apply (auto simp: union_ac)
-done
+lemma multiset_of_merge [simp]:
+  "multiset_of (merge xs ys) = multiset_of xs + multiset_of ys"
+  by (induct xs ys rule: merge.induct) (simp_all add: ac_simps)

-lemma set_merge[simp]: "set (merge xs ys) = set xs \<union> set ys"
-apply(induct xs ys rule: merge.induct)
-apply auto
-done
+lemma set_merge [simp]:
+  "set (merge xs ys) = set xs \<union> set ys"
+  by (induct xs ys rule: merge.induct) auto

-lemma sorted_merge[simp]:
-     "sorted (op \<le>) (merge xs ys) = (sorted (op \<le>) xs & sorted (op \<le>) ys)"
-apply(induct xs ys rule: merge.induct)
-apply(blast intro: order_trans)
-done
+lemma sorted_merge [simp]:
+  "sorted (merge xs ys) \<longleftrightarrow> sorted xs \<and> sorted ys"
+  by (induct xs ys rule: merge.induct) (auto simp add: ball_Un not_le less_le sorted_Cons)

fun msort :: "'a list \<Rightarrow> 'a list"
where
@@ -44,16 +38,19 @@
| "msort xs = merge (msort (take (size xs div 2) xs))
(msort (drop (size xs div 2) xs))"

-theorem sorted_msort: "sorted (op \<le>) (msort xs)"
-by (induct xs rule: msort.induct) simp_all
+lemma sorted_msort:
+  "sorted (msort xs)"
+  by (induct xs rule: msort.induct) simp_all

-theorem multiset_of_msort: "multiset_of (msort xs) = multiset_of xs"
-apply (induct xs rule: msort.induct)
-  apply simp_all
-apply (metis append_take_drop_id drop_Suc_Cons multiset_of.simps(2) multiset_of_append take_Suc_Cons)
-done
+lemma multiset_of_msort:
+  "multiset_of (msort xs) = multiset_of xs"
+  by (induct xs rule: msort.induct)
+    (simp_all, metis append_take_drop_id drop_Suc_Cons multiset_of.simps(2) multiset_of_append take_Suc_Cons)
+
+theorem msort_sort:
+  "sort = msort"
+  by (rule ext, rule properties_for_sort) (fact multiset_of_msort sorted_msort)+

end

-
end```