# HG changeset patch
# User nipkow
# Date 1526214768 -7200
# Node ID 2053ff42214bb3867a4975159b531c17a6a49018
# Parent  efce008331f606ff9893c7b3841f6d4369748382
tuned

diff -r efce008331f6 -r 2053ff42214b src/HOL/Data_Structures/Sorting.thy
--- a/src/HOL/Data_Structures/Sorting.thy	Sun May 13 13:43:34 2018 +0200
+++ b/src/HOL/Data_Structures/Sorting.thy	Sun May 13 14:32:48 2018 +0200
@@ -292,23 +292,23 @@
 
 fun c_merge_adj :: "('a::linorder) list list \<Rightarrow> nat" where
 "c_merge_adj [] = 0" |
-"c_merge_adj [x] = 0" |
-"c_merge_adj (x # y # zs) = c_merge x y + c_merge_adj zs"
+"c_merge_adj [xs] = 0" |
+"c_merge_adj (xs # ys # zss) = c_merge xs ys + c_merge_adj zss"
 
 fun c_merge_all :: "('a::linorder) list list \<Rightarrow> nat" where
 "c_merge_all [] = undefined" |
-"c_merge_all [x] = 0" |
-"c_merge_all xs = c_merge_adj xs + c_merge_all (merge_adj xs)"
+"c_merge_all [xs] = 0" |
+"c_merge_all xss = c_merge_adj xss + c_merge_all (merge_adj xss)"
 
 definition c_msort_bu :: "('a::linorder) list \<Rightarrow> nat" where
 "c_msort_bu xs = (if xs = [] then 0 else c_merge_all (map (\<lambda>x. [x]) xs))"
 
 lemma length_merge_adj:
-  "\<lbrakk> even(length xs); \<forall>x \<in> set xs. length x = m \<rbrakk> \<Longrightarrow> \<forall>x \<in> set (merge_adj xs). length x = 2*m"
-by(induction xs rule: merge_adj.induct) (auto simp: length_merge)
+  "\<lbrakk> even(length xss); \<forall>x \<in> set xss. length x = m \<rbrakk> \<Longrightarrow> \<forall>xs \<in> set (merge_adj xss). length xs = 2*m"
+by(induction xss rule: merge_adj.induct) (auto simp: length_merge)
 
-lemma c_merge_adj: "\<forall>x \<in> set xs. length x = m \<Longrightarrow> c_merge_adj xs \<le> m * length xs"
-proof(induction xs rule: c_merge_adj.induct)
+lemma c_merge_adj: "\<forall>xs \<in> set xss. length xs = m \<Longrightarrow> c_merge_adj xss \<le> m * length xss"
+proof(induction xss rule: c_merge_adj.induct)
   case 1 thus ?case by simp
 next
   case 2 thus ?case by simp
@@ -316,9 +316,9 @@
   case (3 x y) thus ?case using c_merge_ub[of x y] by (simp add: algebra_simps)
 qed
 
-lemma c_merge_all: "\<lbrakk> \<forall>x \<in> set xs. length x = m; length xs = 2^k \<rbrakk>
-  \<Longrightarrow> c_merge_all xs \<le> m * k * 2^k"
-proof (induction xs arbitrary: k m rule: c_merge_all.induct)
+lemma c_merge_all: "\<lbrakk> \<forall>xs \<in> set xss. length xs = m; length xss = 2^k \<rbrakk>
+  \<Longrightarrow> c_merge_all xss \<le> m * k * 2^k"
+proof (induction xss arbitrary: k m rule: c_merge_all.induct)
   case 1 thus ?case by simp
 next
   case 2 thus ?case by simp