src/HOL/Data_Structures/Sorting.thy

 "c_merge_adj [] = 0" |
 "c_merge_adj [x] = 0" |
 "c_merge_adj (x # y # zs) = c_merge x y + c_merge_adj zs"
 
 fun c_merge_all :: "('a::linorder) list list \<Rightarrow> real" where
-"c_merge_all [] = 0" |
+"c_merge_all [] = undefined" |
 "c_merge_all [x] = 0" |
 "c_merge_all xs = c_merge_adj xs + c_merge_all (merge_adj xs)"
 
 definition c_msort_bu :: "('a::linorder) list \<Rightarrow> real" where
 "c_msort_bu xs = (if xs = [] then 0 else c_merge_all (map (\<lambda>x. [x]) xs))"

   let ?xs2 = "merge_adj ?xs"
   obtain k' where k': "k = Suc k'" using "3.prems"(2)
     by (metis length_Cons nat.inject nat_power_eq_Suc_0_iff nat.exhaust)
   have "even (length xs)" using "3.prems"(2) even_Suc_Suc_iff by fastforce
   from "3.prems"(1) length_merge_adj[OF this]
-  have 2: "\<forall>x \<in> set(merge_adj ?xs). length x = 2*m" by(auto simp: length_merge)
-  have 3: "length ?xs2 = 2 ^ k'" using "3.prems"(2) k' by auto
-  have 4: "length ?xs div 2 = 2 ^ k'"
-    using "3.prems"(2) k' by(simp add: power_eq_if[of 2 k] split: if_splits)
+  have *: "\<forall>x \<in> set(merge_adj ?xs). length x = 2*m" by(auto simp: length_merge)
+  have **: "length ?xs2 = 2 ^ k'" using "3.prems"(2) k' by auto
   have "c_merge_all ?xs = c_merge_adj ?xs + c_merge_all ?xs2" by simp
   also have "\<dots> \<le> m * 2^k + c_merge_all ?xs2"
     using "3.prems"(2) c_merge_adj[OF "3.prems"(1)] by (auto simp: algebra_simps)
   also have "\<dots> \<le> m * 2^k + (2*m) * k' * 2^k'"
-    using "3.IH"[OF 2 3] by simp
+    using "3.IH"[OF * **] by simp
   also have "\<dots> = m * k * 2^k"
     using k' by (simp add: algebra_simps)
   finally show ?case .
 qed