src/HOL/ex/MergeSort.thy

 *)
 
 header{*Merge Sort*}
 
 theory MergeSort
-imports Sorting
+imports Multiset
 begin
 
 context linorder
 begin
 

   "merge (x#xs) (y#ys) =
          (if x \<le> y then x # merge xs (y#ys) else y # merge (x#xs) ys)"
 | "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(simp_all add: ball_Un not_le less_le)
-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
   "msort [] = []"
 | "msort [x] = [x]"
 | "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