--- a/src/HOL/Data_Structures/Sorting.thy Fri May 04 15:59:21 2018 +0200
+++ b/src/HOL/Data_Structures/Sorting.thy Fri May 04 18:41:11 2018 +0200
@@ -312,7 +312,7 @@
"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)"
@@ -347,15 +347,13 @@
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 .