diff --git a/src/HOL/Data_Structures/Array_Braun.thy b/src/HOL/Data_Structures/Array_Braun.thy
--- a/src/HOL/Data_Structures/Array_Braun.thy
+++ b/src/HOL/Data_Structures/Array_Braun.thy
@@ -1,644 +1,668 @@
 (* Author: Tobias Nipkow, with contributions by Thomas Sewell *)
 
 section "Arrays via Braun Trees"
 
 theory Array_Braun
 imports
   Array_Specs
   Braun_Tree
 begin
 
 subsection "Array"
 
 fun lookup1 :: "'a tree \<Rightarrow> nat \<Rightarrow> 'a" where
 "lookup1 (Node l x r) n = (if n=1 then x else lookup1 (if even n then l else r) (n div 2))"
 
 fun update1 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
 "update1 n x Leaf = Node Leaf x Leaf" |
 "update1 n x (Node l a r) =
   (if n=1 then Node l x r else
    if even n then Node (update1 (n div 2) x l) a r
             else Node l a (update1 (n div 2) x r))"
 
 fun adds :: "'a list \<Rightarrow> nat \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
 "adds [] n t = t" |
 "adds (x#xs) n t = adds xs (n+1) (update1 (n+1) x t)"
 
 fun list :: "'a tree \<Rightarrow> 'a list" where
 "list Leaf = []" |
 "list (Node l x r) = x # splice (list l) (list r)"
 
 
 subsubsection "Functional Correctness"
 
 lemma size_list: "size(list t) = size t"
 by(induction t)(auto)
 
 lemma minus1_div2: "(n - Suc 0) div 2 = (if odd n then n div 2 else n div 2 - 1)"
 by auto arith
 
 lemma nth_splice: "\<lbrakk> n < size xs + size ys;  size ys \<le> size xs;  size xs \<le> size ys + 1 \<rbrakk>
   \<Longrightarrow> splice xs ys ! n = (if even n then xs else ys) ! (n div 2)"
 apply(induction xs ys arbitrary: n rule: splice.induct)
 apply (auto simp: nth_Cons' minus1_div2)
 done
 
 lemma div2_in_bounds:
   "\<lbrakk> braun (Node l x r); n \<in> {1..size(Node l x r)}; n > 1 \<rbrakk> \<Longrightarrow>
    (odd n \<longrightarrow> n div 2 \<in> {1..size r}) \<and> (even n \<longrightarrow> n div 2 \<in> {1..size l})"
 by auto arith
 
 declare upt_Suc[simp del]
 
 
 paragraph \<open>\<^const>\<open>lookup1\<close>\<close>
 
 lemma nth_list_lookup1: "\<lbrakk>braun t; i < size t\<rbrakk> \<Longrightarrow> list t ! i = lookup1 t (i+1)"
 proof(induction t arbitrary: i)
   case Leaf thus ?case by simp
 next
   case Node
   thus ?case using div2_in_bounds[OF Node.prems(1), of "i+1"]
     by (auto simp: nth_splice minus1_div2 size_list)
 qed
 
 lemma list_eq_map_lookup1: "braun t \<Longrightarrow> list t = map (lookup1 t) [1..<size t + 1]"
 by(auto simp add: list_eq_iff_nth_eq size_list nth_list_lookup1)
 
 
 paragraph \<open>\<^const>\<open>update1\<close>\<close>
 
 lemma size_update1: "\<lbrakk> braun t;  n \<in> {1.. size t} \<rbrakk> \<Longrightarrow> size(update1 n x t) = size t"
 proof(induction t arbitrary: n)
   case Leaf thus ?case by simp
 next
   case Node thus ?case using div2_in_bounds[OF Node.prems] by simp
 qed
 
 lemma braun_update1: "\<lbrakk>braun t;  n \<in> {1.. size t} \<rbrakk> \<Longrightarrow> braun(update1 n x t)"
 proof(induction t arbitrary: n)
   case Leaf thus ?case by simp
 next
   case Node thus ?case
     using div2_in_bounds[OF Node.prems] by (simp add: size_update1)
 qed
 
 lemma lookup1_update1: "\<lbrakk> braun t;  n \<in> {1.. size t} \<rbrakk> \<Longrightarrow>
   lookup1 (update1 n x t) m = (if n=m then x else lookup1 t m)"
 proof(induction t arbitrary: m n)
   case Leaf
   then show ?case by simp
 next
   have aux: "\<lbrakk> odd n; odd m \<rbrakk> \<Longrightarrow> n div 2 = (m::nat) div 2 \<longleftrightarrow> m=n" for m n
     using odd_two_times_div_two_succ by fastforce
   case Node
   thus ?case using div2_in_bounds[OF Node.prems] by (auto simp: aux)
 qed
 
 lemma list_update1: "\<lbrakk> braun t;  n \<in> {1.. size t} \<rbrakk> \<Longrightarrow> list(update1 n x t) = (list t)[n-1 := x]"
 by(auto simp add: list_eq_map_lookup1 list_eq_iff_nth_eq lookup1_update1 size_update1 braun_update1)
 
 text \<open>A second proof of @{thm list_update1}:\<close>
 
 lemma diff1_eq_iff: "n > 0 \<Longrightarrow> n - Suc 0 = m \<longleftrightarrow> n = m+1"
 by arith
 
 lemma list_update_splice:
   "\<lbrakk> n < size xs + size ys;  size ys \<le> size xs;  size xs \<le> size ys + 1 \<rbrakk> \<Longrightarrow>
   (splice xs ys) [n := x] =
   (if even n then splice (xs[n div 2 := x]) ys else splice xs (ys[n div 2 := x]))"
 by(induction xs ys arbitrary: n rule: splice.induct) (auto split: nat.split)
 
 lemma list_update2: "\<lbrakk> braun t;  n \<in> {1.. size t} \<rbrakk> \<Longrightarrow> list(update1 n x t) = (list t)[n-1 := x]"
 proof(induction t arbitrary: n)
   case Leaf thus ?case by simp
 next
   case (Node l a r) thus ?case using div2_in_bounds[OF Node.prems]
     by(auto simp: list_update_splice diff1_eq_iff size_list split: nat.split)
 qed
 
 
 paragraph \<open>\<^const>\<open>adds\<close>\<close>
 
 lemma splice_last: shows
   "size ys \<le> size xs \<Longrightarrow> splice (xs @ [x]) ys = splice xs ys @ [x]"
 and "size ys+1 \<ge> size xs \<Longrightarrow> splice xs (ys @ [y]) = splice xs ys @ [y]"
 by(induction xs ys arbitrary: x y rule: splice.induct) (auto)
 
 lemma list_add_hi: "braun t \<Longrightarrow> list(update1 (Suc(size t)) x t) = list t @ [x]"
 by(induction t)(auto simp: splice_last size_list)
 
 lemma size_add_hi: "braun t \<Longrightarrow> m = size t \<Longrightarrow> size(update1 (Suc m) x t) = size t + 1"
 by(induction t arbitrary: m)(auto)
 
 lemma braun_add_hi: "braun t \<Longrightarrow> braun(update1 (Suc(size t)) x t)"
 by(induction t)(auto simp: size_add_hi)
 
 lemma size_braun_adds:
   "\<lbrakk> braun t; size t = n \<rbrakk> \<Longrightarrow> size(adds xs n t) = size t + length xs \<and> braun (adds xs n t)"
 by(induction xs arbitrary: t n)(auto simp: braun_add_hi size_add_hi)
 
 lemma list_adds: "\<lbrakk> braun t; size t = n \<rbrakk> \<Longrightarrow> list(adds xs n t) = list t @ xs"
 by(induction xs arbitrary: t n)(auto simp: size_braun_adds list_add_hi size_add_hi braun_add_hi)
 
 
 subsubsection "Array Implementation"
 
 interpretation A: Array
 where lookup = "\<lambda>(t,l) n. lookup1 t (n+1)"
 and update = "\<lambda>n x (t,l). (update1 (n+1) x t, l)"
 and len = "\<lambda>(t,l). l"
 and array = "\<lambda>xs. (adds xs 0 Leaf, length xs)"
 and invar = "\<lambda>(t,l). braun t \<and> l = size t"
 and list = "\<lambda>(t,l). list t"
 proof (standard, goal_cases)
   case 1 thus ?case by (simp add: nth_list_lookup1 split: prod.splits)
 next
   case 2 thus ?case by (simp add: list_update1 split: prod.splits)
 next
   case 3 thus ?case by (simp add: size_list split: prod.splits)
 next
   case 4 thus ?case by (simp add: list_adds)
 next
   case 5 thus ?case by (simp add: braun_update1 size_update1 split: prod.splits)
 next
   case 6 thus ?case by (simp add: size_braun_adds split: prod.splits)
 qed
 
 
 subsection "Flexible Array"
 
 fun add_lo where
 "add_lo x Leaf = Node Leaf x Leaf" |
 "add_lo x (Node l a r) = Node (add_lo a r) x l"
 
 fun merge where
 "merge Leaf r = r" |
 "merge (Node l a r) rr = Node rr a (merge l r)"
 
 fun del_lo where
 "del_lo Leaf = Leaf" |
 "del_lo (Node l a r) = merge l r"
 
 fun del_hi :: "nat \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
 "del_hi n Leaf = Leaf" |
 "del_hi n (Node l x r) =
   (if n = 1 then Leaf
    else if even n
        then Node (del_hi (n div 2) l) x r
        else Node l x (del_hi (n div 2) r))"
 
 
 subsubsection "Functional Correctness"
 
 
 paragraph \<open>\<^const>\<open>add_lo\<close>\<close>
 
 lemma list_add_lo: "braun t \<Longrightarrow> list (add_lo a t) = a # list t"
 by(induction t arbitrary: a) auto
 
 lemma braun_add_lo: "braun t \<Longrightarrow> braun(add_lo x t)"
 by(induction t arbitrary: x) (auto simp add: list_add_lo simp flip: size_list)
 
 
 paragraph \<open>\<^const>\<open>del_lo\<close>\<close>
 
 lemma list_merge: "braun (Node l x r) \<Longrightarrow> list(merge l r) = splice (list l) (list r)"
 by (induction l r rule: merge.induct) auto
 
 lemma braun_merge: "braun (Node l x r) \<Longrightarrow> braun(merge l r)"
 by (induction l r rule: merge.induct)(auto simp add: list_merge simp flip: size_list)
 
 lemma list_del_lo: "braun t \<Longrightarrow> list(del_lo t) = tl (list t)"
 by (cases t) (simp_all add: list_merge)
 
 lemma braun_del_lo: "braun t \<Longrightarrow> braun(del_lo t)"
 by (cases t) (simp_all add: braun_merge)
 
 
 paragraph \<open>\<^const>\<open>del_hi\<close>\<close>
 
 lemma list_Nil_iff: "list t = [] \<longleftrightarrow> t = Leaf"
 by(cases t) simp_all
 
 lemma butlast_splice: "butlast (splice xs ys) =
   (if size xs > size ys then splice (butlast xs) ys else splice xs (butlast ys))"
 by(induction xs ys rule: splice.induct) (auto)
 
 lemma list_del_hi: "braun t \<Longrightarrow> size t = st \<Longrightarrow> list(del_hi st t) = butlast(list t)"
 apply(induction t arbitrary: st)
 by(auto simp: list_Nil_iff size_list butlast_splice)
 
 lemma braun_del_hi: "braun t \<Longrightarrow> size t = st \<Longrightarrow> braun(del_hi st t)"
 apply(induction t arbitrary: st)
 by(auto simp: list_del_hi simp flip: size_list)
 
 
 subsubsection "Flexible Array Implementation"
 
 interpretation AF: Array_Flex
 where lookup = "\<lambda>(t,l) n. lookup1 t (n+1)"
 and update = "\<lambda>n x (t,l). (update1 (n+1) x t, l)"
 and len = "\<lambda>(t,l). l"
 and array = "\<lambda>xs. (adds xs 0 Leaf, length xs)"
 and invar = "\<lambda>(t,l). braun t \<and> l = size t"
 and list = "\<lambda>(t,l). list t"
 and add_lo = "\<lambda>x (t,l). (add_lo x t, l+1)"
 and del_lo = "\<lambda>(t,l). (del_lo t, l-1)"
 and add_hi = "\<lambda>x (t,l). (update1 (Suc l) x t, l+1)"
 and del_hi = "\<lambda>(t,l). (del_hi l t, l-1)"
 proof (standard, goal_cases)
   case 1 thus ?case by (simp add: list_add_lo split: prod.splits)
 next
   case 2 thus ?case by (simp add: list_del_lo split: prod.splits)
 next
   case 3 thus ?case by (simp add: list_add_hi braun_add_hi split: prod.splits)
 next
   case 4 thus ?case by (simp add: list_del_hi split: prod.splits)
 next
   case 5 thus ?case by (simp add: braun_add_lo list_add_lo flip: size_list split: prod.splits)
 next
   case 6 thus ?case by (simp add: braun_del_lo list_del_lo flip: size_list split: prod.splits)
 next
   case 7 thus ?case by (simp add: size_add_hi braun_add_hi split: prod.splits)
 next
   case 8 thus ?case by (simp add: braun_del_hi list_del_hi flip: size_list split: prod.splits)
 qed
 
 
 subsection "Faster"
 
 
 subsubsection \<open>Size\<close>
 
 fun diff :: "'a tree \<Rightarrow> nat \<Rightarrow> nat" where
 "diff Leaf _ = 0" |
 "diff (Node l x r) n = (if n=0 then 1 else if even n then diff r (n div 2 - 1) else diff l (n div 2))"
 
 fun size_fast :: "'a tree \<Rightarrow> nat" where
 "size_fast Leaf = 0" |
 "size_fast (Node l x r) = (let n = size_fast r in 1 + 2*n + diff l n)"
 
 lemma diff: "braun t \<Longrightarrow> size t : {n, n + 1} \<Longrightarrow> diff t n = size t - n"
 by(induction t arbitrary: n) auto
 
 lemma size_fast: "braun t \<Longrightarrow> size_fast t = size t"
 by(induction t) (auto simp add: Let_def diff)
 
 
 subsubsection \<open>Initialization with 1 element\<close>
 
 fun braun_of_naive :: "'a \<Rightarrow> nat \<Rightarrow> 'a tree" where
 "braun_of_naive x n = (if n=0 then Leaf
   else let m = (n-1) div 2
        in if odd n then Node (braun_of_naive x m) x (braun_of_naive x m)
        else Node (braun_of_naive x (m + 1)) x (braun_of_naive x m))"
 
 fun braun2_of :: "'a \<Rightarrow> nat \<Rightarrow> 'a tree * 'a tree" where
 "braun2_of x n = (if n = 0 then (Leaf, Node Leaf x Leaf)
   else let (s,t) = braun2_of x ((n-1) div 2)
        in if odd n then (Node s x s, Node t x s) else (Node t x s, Node t x t))"
 
 definition braun_of :: "'a \<Rightarrow> nat \<Rightarrow> 'a tree" where
 "braun_of x n = fst (braun2_of x n)"
 
 declare braun2_of.simps [simp del]
 
 lemma braun2_of_size_braun: "braun2_of x n = (s,t) \<Longrightarrow> size s = n \<and> size t = n+1 \<and> braun s \<and> braun t"
 proof(induction x n arbitrary: s t rule: braun2_of.induct)
   case (1 x n)
   then show ?case
     by (auto simp: braun2_of.simps[of x n] split: prod.splits if_splits) presburger+
 qed
 
 lemma braun2_of_replicate:
   "braun2_of x n = (s,t) \<Longrightarrow> list s = replicate n x \<and> list t = replicate (n+1) x"
 proof(induction x n arbitrary: s t rule: braun2_of.induct)
   case (1 x n)
   have "x # replicate m x = replicate (m+1) x" for m by simp
   with 1 show ?case
     apply (auto simp: braun2_of.simps[of x n] replicate.simps(2)[of 0 x]
         simp del: replicate.simps(2) split: prod.splits if_splits)
     by presburger+
 qed
 
 corollary braun_braun_of: "braun(braun_of x n)"
 unfolding braun_of_def by (metis eq_fst_iff braun2_of_size_braun)
 
 corollary list_braun_of: "list(braun_of x n) = replicate n x"
 unfolding braun_of_def by (metis eq_fst_iff braun2_of_replicate)
 
 
 subsubsection "Proof Infrastructure"
 
 text \<open>Originally due to Thomas Sewell.\<close>
 
 paragraph \<open>\<open>take_nths\<close>\<close>
 
 fun take_nths :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
 "take_nths i k [] = []" |
 "take_nths i k (x # xs) = (if i = 0 then x # take_nths (2^k - 1) k xs
   else take_nths (i - 1) k xs)"
 
+text \<open>This is the more concise definition but seems to complicate the proofs:\<close>
+
+lemma take_nths_eq_nths: "take_nths i k xs = nths xs (\<Union>n. {n*2^k + i})"
+proof(induction xs arbitrary: i)
+  case Nil
+  then show ?case by simp
+next
+  case (Cons x xs)
+  show ?case
+  proof cases
+    assume [simp]: "i = 0"
+    have "(\<Union>n. {(n+1) * 2 ^ k - 1}) = {m. \<exists>n. Suc m = n * 2 ^ k}"
+      apply (auto simp del: mult_Suc)
+      by (metis diff_Suc_Suc diff_zero mult_eq_0_iff not0_implies_Suc)
+    thus ?thesis by (simp add: Cons.IH ac_simps nths_Cons)
+  next
+    assume [arith]: "i \<noteq> 0"
+    have "(\<Union>n. {n * 2 ^ k + i - 1}) = {m. \<exists>n. Suc m = n * 2 ^ k + i}"
+      apply auto
+      by (metis diff_Suc_Suc diff_zero)
+    thus ?thesis by (simp add: Cons.IH nths_Cons)
+  qed
+qed
+
 lemma take_nths_drop:
   "take_nths i k (drop j xs) = take_nths (i + j) k xs"
 by (induct xs arbitrary: i j; simp add: drop_Cons split: nat.split)
 
 lemma take_nths_00:
   "take_nths 0 0 xs = xs"
 by (induct xs; simp)
 
 lemma splice_take_nths:
   "splice (take_nths 0 (Suc 0) xs) (take_nths (Suc 0) (Suc 0) xs) = xs"
 by (induct xs; simp)
 
 lemma take_nths_take_nths:
   "take_nths i m (take_nths j n xs) = take_nths ((i * 2^n) + j) (m + n) xs"
 by (induct xs arbitrary: i j; simp add: algebra_simps power_add)
 
 lemma take_nths_empty:
   "(take_nths i k xs = []) = (length xs \<le> i)"
 by (induction xs arbitrary: i k) auto
 
 lemma hd_take_nths:
   "i < length xs \<Longrightarrow> hd(take_nths i k xs) = xs ! i"
 by (induction xs arbitrary: i k) auto
 
 lemma take_nths_01_splice:
   "\<lbrakk> length xs = length ys \<or> length xs = length ys + 1 \<rbrakk> \<Longrightarrow>
    take_nths 0 (Suc 0) (splice xs ys) = xs \<and>
    take_nths (Suc 0) (Suc 0) (splice xs ys) = ys"
 by (induct xs arbitrary: ys; case_tac ys; simp)
 
 lemma length_take_nths_00:
   "length (take_nths 0 (Suc 0) xs) = length (take_nths (Suc 0) (Suc 0) xs) \<or>
    length (take_nths 0 (Suc 0) xs) = length (take_nths (Suc 0) (Suc 0) xs) + 1"
 by (induct xs) auto
 
 
 paragraph \<open>\<open>braun_list\<close>\<close>
 
 fun braun_list :: "'a tree \<Rightarrow> 'a list \<Rightarrow> bool" where
 "braun_list Leaf xs = (xs = [])" |
 "braun_list (Node l x r) xs = (xs \<noteq> [] \<and> x = hd xs \<and>
     braun_list l (take_nths 1 1 xs) \<and>
     braun_list r (take_nths 2 1 xs))"
 
 lemma braun_list_eq:
   "braun_list t xs = (braun t \<and> xs = list t)"
 proof (induct t arbitrary: xs)
   case Leaf
   show ?case by simp
 next
   case Node
   show ?case
     using length_take_nths_00[of xs] splice_take_nths[of xs]
     by (auto simp: neq_Nil_conv Node.hyps size_list[symmetric] take_nths_01_splice)
 qed
 
 
 subsubsection \<open>Converting a list of elements into a Braun tree\<close>
 
 fun nodes :: "'a tree list \<Rightarrow> 'a list \<Rightarrow> 'a tree list \<Rightarrow> 'a tree list" where
 "nodes (l#ls) (x#xs) (r#rs) = Node l x r # nodes ls xs rs" |
 "nodes (l#ls) (x#xs) [] = Node l x Leaf # nodes ls xs []" |
 "nodes [] (x#xs) (r#rs) = Node Leaf x r # nodes [] xs rs" |
 "nodes [] (x#xs) [] = Node Leaf x Leaf # nodes [] xs []" |
 "nodes ls [] rs = []"
 
 fun brauns :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a tree list" where
 "brauns k xs = (if xs = [] then [] else
    let ys = take (2^k) xs;
        zs = drop (2^k) xs;
        ts = brauns (k+1) zs
    in nodes ts ys (drop (2^k) ts))"
 
 declare brauns.simps[simp del]
 
 definition brauns1 :: "'a list \<Rightarrow> 'a tree" where
 "brauns1 xs = (if xs = [] then Leaf else brauns 0 xs ! 0)"
 
 fun t_brauns :: "nat \<Rightarrow> 'a list \<Rightarrow> nat" where
 "t_brauns k xs = (if xs = [] then 0 else
    let ys = take (2^k) xs;
        zs = drop (2^k) xs;
        ts = brauns (k+1) zs
    in 4 * min (2^k) (length xs) + t_brauns (k+1) zs)"
 
 
 paragraph "Functional correctness"
 
 text \<open>The proof is originally due to Thomas Sewell.\<close>
 
 lemma length_nodes:
   "length (nodes ls xs rs) = length xs"
 by (induct ls xs rs rule: nodes.induct; simp)
 
 lemma nth_nodes:
   "i < length xs \<Longrightarrow> nodes ls xs rs ! i =
   Node (if i < length ls then ls ! i else Leaf) (xs ! i)
     (if i < length rs then rs ! i else Leaf)"
 by (induct ls xs rs arbitrary: i rule: nodes.induct;
     simp add: nth_Cons split: nat.split)
 
 theorem length_brauns:
   "length (brauns k xs) = min (length xs) (2 ^ k)"
 proof (induct xs arbitrary: k rule: measure_induct_rule[where f=length])
   case (less xs) thus ?case by (simp add: brauns.simps[of k xs] Let_def length_nodes)
 qed
 
 theorem brauns_correct:
   "i < min (length xs) (2 ^ k) \<Longrightarrow> braun_list (brauns k xs ! i) (take_nths i k xs)"
 proof (induct xs arbitrary: i k rule: measure_induct_rule[where f=length])
   case (less xs)
   have "xs \<noteq> []" using less.prems by auto
   let ?zs = "drop (2^k) xs"
   let ?ts = "brauns (Suc k) ?zs"
   from less.hyps[of ?zs _ "Suc k"]
   have IH: "\<lbrakk> j = i + 2 ^ k;  i < min (length ?zs) (2 ^ (k+1)) \<rbrakk> \<Longrightarrow>
     braun_list (?ts ! i) (take_nths j (Suc k) xs)" for i j
     using \<open>xs \<noteq> []\<close> by (simp add: take_nths_drop)
   show ?case
     using less.prems
     by (auto simp: brauns.simps[of k xs] Let_def nth_nodes take_nths_take_nths
                    IH take_nths_empty hd_take_nths length_brauns)
 qed
 
 corollary brauns1_correct:
   "braun (brauns1 xs) \<and> list (brauns1 xs) = xs"
 using brauns_correct[of 0 xs 0]
 by (simp add: brauns1_def braun_list_eq take_nths_00)
 
 
 paragraph "Running Time Analysis"
 
 theorem t_brauns:
   "t_brauns k xs = 4 * length xs"
 proof (induction xs arbitrary: k rule: measure_induct_rule[where f = length])
   case (less xs)
   show ?case
   proof cases
     assume "xs = []"
     thus ?thesis by(simp add: Let_def)
   next
     assume "xs \<noteq> []"
     let ?zs = "drop (2^k) xs"
     have "t_brauns k xs = t_brauns (k+1) ?zs + 4 * min (2^k) (length xs)"
      using \<open>xs \<noteq> []\<close> by(simp add: Let_def)
     also have "\<dots> = 4 * length ?zs + 4 * min (2^k) (length xs)"
       using less[of ?zs "k+1"] \<open>xs \<noteq> []\<close>
       by (simp)
     also have "\<dots> = 4 * length xs"
       by(simp)
     finally show ?case .
   qed
 qed
 
 
 subsubsection \<open>Converting a Braun Tree into a List of Elements\<close>
 
 text \<open>The code and the proof are originally due to Thomas Sewell (except running time).\<close>
 
 function list_fast_rec :: "'a tree list \<Rightarrow> 'a list" where
 "list_fast_rec ts = (let us = filter (\<lambda>t. t \<noteq> Leaf) ts in
   if us = [] then [] else
   map value us @ list_fast_rec (map left us @ map right us))"
 by (pat_completeness, auto)
 
 lemma list_fast_rec_term1: "ts \<noteq> [] \<Longrightarrow> Leaf \<notin> set ts \<Longrightarrow>
   sum_list (map (size o left) ts) + sum_list (map (size o right) ts) < sum_list (map size ts)"
   apply (clarsimp simp: sum_list_addf[symmetric] sum_list_map_filter')
   apply (rule sum_list_strict_mono; clarsimp?)
   apply (case_tac x; simp)
   done
 
 lemma list_fast_rec_term: "us \<noteq> [] \<Longrightarrow> us = filter (\<lambda>t. t \<noteq> \<langle>\<rangle>) ts \<Longrightarrow>
   sum_list (map (size o left) us) + sum_list (map (size o right) us) < sum_list (map size ts)"
   apply (rule order_less_le_trans, rule list_fast_rec_term1, simp_all)
   apply (rule sum_list_filter_le_nat)
   done
 
 termination
   apply (relation "measure (sum_list o map size)")
    apply simp
   apply (simp add: list_fast_rec_term)
   done
 
 declare list_fast_rec.simps[simp del]
 
 definition list_fast :: "'a tree \<Rightarrow> 'a list" where
 "list_fast t = list_fast_rec [t]"
 
 function t_list_fast_rec :: "'a tree list \<Rightarrow> nat" where
 "t_list_fast_rec ts = (let us = filter (\<lambda>t. t \<noteq> Leaf) ts
   in length ts + (if us = [] then 0 else
   5 * length us + t_list_fast_rec (map left us @ map right us)))"
 by (pat_completeness, auto)
 
 termination
   apply (relation "measure (sum_list o map size)")
    apply simp
   apply (simp add: list_fast_rec_term)
   done
 
 declare t_list_fast_rec.simps[simp del]
 
 paragraph "Functional Correctness"
 
 lemma list_fast_rec_all_Leaf:
   "\<forall>t \<in> set ts. t = Leaf \<Longrightarrow> list_fast_rec ts = []"
 by (simp add: filter_empty_conv list_fast_rec.simps)
 
 lemma take_nths_eq_single:
   "length xs - i < 2^n \<Longrightarrow> take_nths i n xs = take 1 (drop i xs)"
 by (induction xs arbitrary: i n; simp add: drop_Cons')
 
 lemma braun_list_Nil:
   "braun_list t [] = (t = Leaf)"
 by (cases t; simp)
 
 lemma braun_list_not_Nil: "xs \<noteq> [] \<Longrightarrow>
   braun_list t xs =
  (\<exists>l x r. t = Node l x r \<and> x = hd xs \<and>
     braun_list l (take_nths 1 1 xs) \<and>
     braun_list r (take_nths 2 1 xs))"
 by(cases t; simp)
 
 theorem list_fast_rec_correct:
   "\<lbrakk> length ts = 2 ^ k; \<forall>i < 2 ^ k. braun_list (ts ! i) (take_nths i k xs) \<rbrakk>
     \<Longrightarrow> list_fast_rec ts = xs"
 proof (induct xs arbitrary: k ts rule: measure_induct_rule[where f=length])
   case (less xs)
   show ?case
   proof (cases "length xs < 2 ^ k")
     case True
     from less.prems True have filter:
       "\<exists>n. ts = map (\<lambda>x. Node Leaf x Leaf) xs @ replicate n Leaf"
       apply (rule_tac x="length ts - length xs" in exI)
       apply (clarsimp simp: list_eq_iff_nth_eq)
       apply(auto simp: nth_append braun_list_not_Nil take_nths_eq_single braun_list_Nil hd_drop_conv_nth)
       done
     thus ?thesis
       by (clarsimp simp: list_fast_rec.simps[of ts] o_def list_fast_rec_all_Leaf Let_def)
   next
     case False
     with less.prems(2) have *:
       "\<forall>i < 2 ^ k. ts ! i \<noteq> Leaf
          \<and> value (ts ! i) = xs ! i
          \<and> braun_list (left (ts ! i)) (take_nths (i + 2 ^ k) (Suc k) xs)
          \<and> (\<forall>ys. ys = take_nths (i + 2 * 2 ^ k) (Suc k) xs
                  \<longrightarrow> braun_list (right (ts ! i)) ys)"
       by (auto simp: take_nths_empty hd_take_nths braun_list_not_Nil take_nths_take_nths
                      algebra_simps)
     have 1: "map value ts = take (2 ^ k) xs"
       using less.prems(1) False by (simp add: list_eq_iff_nth_eq *)
     have 2: "list_fast_rec (map left ts @ map right ts) = drop (2 ^ k) xs"
       using less.prems(1) False
       by (auto intro!: Nat.diff_less less.hyps[where k= "Suc k"]
                simp: nth_append * take_nths_drop algebra_simps)
     from less.prems(1) False show ?thesis
       by (auto simp: list_fast_rec.simps[of ts] 1 2 Let_def * all_set_conv_all_nth)
   qed
 qed
 
 corollary list_fast_correct:
   "braun t \<Longrightarrow> list_fast t = list t"
 by (simp add: list_fast_def take_nths_00 braun_list_eq list_fast_rec_correct[where k=0])
 
 paragraph "Running Time Analysis"
 
 lemma sum_tree_list_children: "\<forall>t \<in> set ts. t \<noteq> Leaf \<Longrightarrow>
   (\<Sum>t\<leftarrow>ts. k * size t) = (\<Sum>t \<leftarrow> map left ts @ map right ts. k * size t) + k * length ts"
 by(induction ts)(auto simp add: neq_Leaf_iff algebra_simps)
 
 theorem t_list_fast_rec_ub:
   "t_list_fast_rec ts \<le> sum_list (map (\<lambda>t. 7*size t + 1) ts)"
 proof (induction ts rule: measure_induct_rule[where f="sum_list o map size"])
   case (less ts)
   let ?us = "filter (\<lambda>t. t \<noteq> Leaf) ts"
   show ?case
   proof cases
     assume "?us = []"
     thus ?thesis using t_list_fast_rec.simps[of ts]
       by(simp add: Let_def sum_list_Suc)
     next
     assume "?us \<noteq> []"
     let ?children = "map left ?us @ map right ?us"
     have "t_list_fast_rec ts = t_list_fast_rec ?children + 5 * length ?us + length ts"
      using \<open>?us \<noteq> []\<close> t_list_fast_rec.simps[of ts] by(simp add: Let_def)
     also have "\<dots> \<le> (\<Sum>t\<leftarrow>?children. 7 * size t + 1) + 5 * length ?us + length ts"
       using less[of "?children"] list_fast_rec_term[of "?us"] \<open>?us \<noteq> []\<close>
       by (simp)
     also have "\<dots> = (\<Sum>t\<leftarrow>?children. 7*size t) + 7 * length ?us + length ts"
       by(simp add: sum_list_Suc o_def)
     also have "\<dots> = (\<Sum>t\<leftarrow>?us. 7*size t) + length ts"
       by(simp add: sum_tree_list_children)
     also have "\<dots> \<le> (\<Sum>t\<leftarrow>ts. 7*size t) + length ts"
       by(simp add: sum_list_filter_le_nat)
     also have "\<dots> = (\<Sum>t\<leftarrow>ts. 7 * size t + 1)"
       by(simp add: sum_list_Suc)
     finally show ?case .
   qed
 qed
 
 end
\ No newline at end of file