--- a/src/HOL/Data_Structures/Sorting.thy Sun May 13 14:32:48 2018 +0200
+++ b/src/HOL/Data_Structures/Sorting.thy Sun May 13 14:40:40 2018 +0200
@@ -323,17 +323,17 @@
next
case 2 thus ?case by simp
next
- case (3 x y xs)
- let ?xs = "x # y # xs"
- let ?xs2 = "merge_adj ?xs"
+ case (3 xs ys xss)
+ let ?xss = "xs # ys # xss"
+ let ?xss2 = "merge_adj ?xss"
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
+ have "even (length xss)" using "3.prems"(2) even_Suc_Suc_iff by fastforce
from "3.prems"(1) length_merge_adj[OF this]
- 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"
+ have *: "\<forall>x \<in> set(merge_adj ?xss). length x = 2*m" by(auto simp: length_merge)
+ have **: "length ?xss2 = 2 ^ k'" using "3.prems"(2) k' by auto
+ have "c_merge_all ?xss = c_merge_adj ?xss + c_merge_all ?xss2" by simp
+ also have "\<dots> \<le> m * 2^k + c_merge_all ?xss2"
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 * **] by simp