diff --git a/src/HOL/Library/Tree.thy b/src/HOL/Library/Tree.thy
--- a/src/HOL/Library/Tree.thy
+++ b/src/HOL/Library/Tree.thy
@@ -1,451 +1,454 @@
 (* Author: Tobias Nipkow *)
 (* Todo: minimal ipl of balanced trees *)
 
 section \<open>Binary Tree\<close>
 
 theory Tree
 imports Main
 begin
 
 datatype 'a tree =
   Leaf ("\<langle>\<rangle>") |
   Node "'a tree" ("value": 'a) "'a tree" ("(1\<langle>_,/ _,/ _\<rangle>)")
 datatype_compat tree
 
 primrec left :: "'a tree \<Rightarrow> 'a tree" where
 "left (Node l v r) = l" |
 "left Leaf = Leaf"
 
 primrec right :: "'a tree \<Rightarrow> 'a tree" where
 "right (Node l v r) = r" |
 "right Leaf = Leaf"
 
 text\<open>Counting the number of leaves rather than nodes:\<close>
 
 fun size1 :: "'a tree \<Rightarrow> nat" where
 "size1 \<langle>\<rangle> = 1" |
 "size1 \<langle>l, x, r\<rangle> = size1 l + size1 r"
 
 fun subtrees :: "'a tree \<Rightarrow> 'a tree set" where
 "subtrees \<langle>\<rangle> = {\<langle>\<rangle>}" |
 "subtrees (\<langle>l, a, r\<rangle>) = insert \<langle>l, a, r\<rangle> (subtrees l \<union> subtrees r)"
 
 fun mirror :: "'a tree \<Rightarrow> 'a tree" where
 "mirror \<langle>\<rangle> = Leaf" |
 "mirror \<langle>l,x,r\<rangle> = \<langle>mirror r, x, mirror l\<rangle>"
 
 class height = fixes height :: "'a \<Rightarrow> nat"
 
 instantiation tree :: (type)height
 begin
 
 fun height_tree :: "'a tree => nat" where
 "height Leaf = 0" |
 "height (Node l a r) = max (height l) (height r) + 1"
 
 instance ..
 
 end
 
 fun min_height :: "'a tree \<Rightarrow> nat" where
 "min_height Leaf = 0" |
 "min_height (Node l _ r) = min (min_height l) (min_height r) + 1"
 
 fun complete :: "'a tree \<Rightarrow> bool" where
 "complete Leaf = True" |
 "complete (Node l x r) = (complete l \<and> complete r \<and> height l = height r)"
 
 definition balanced :: "'a tree \<Rightarrow> bool" where
 "balanced t = (height t - min_height t \<le> 1)"
 
 text \<open>Weight balanced:\<close>
 fun wbalanced :: "'a tree \<Rightarrow> bool" where
 "wbalanced Leaf = True" |
 "wbalanced (Node l x r) = (abs(int(size l) - int(size r)) \<le> 1 \<and> wbalanced l \<and> wbalanced r)"
 
 text \<open>Internal path length:\<close>
 fun ipl :: "'a tree \<Rightarrow> nat" where
 "ipl Leaf = 0 " |
 "ipl (Node l _ r) = ipl l + size l + ipl r + size r"
 
 fun preorder :: "'a tree \<Rightarrow> 'a list" where
 "preorder \<langle>\<rangle> = []" |
 "preorder \<langle>l, x, r\<rangle> = x # preorder l @ preorder r"
 
 fun inorder :: "'a tree \<Rightarrow> 'a list" where
 "inorder \<langle>\<rangle> = []" |
 "inorder \<langle>l, x, r\<rangle> = inorder l @ [x] @ inorder r"
 
 text\<open>A linear version avoiding append:\<close>
 fun inorder2 :: "'a tree \<Rightarrow> 'a list \<Rightarrow> 'a list" where
 "inorder2 \<langle>\<rangle> xs = xs" |
 "inorder2 \<langle>l, x, r\<rangle> xs = inorder2 l (x # inorder2 r xs)"
 
 fun postorder :: "'a tree \<Rightarrow> 'a list" where
 "postorder \<langle>\<rangle> = []" |
 "postorder \<langle>l, x, r\<rangle> = postorder l @ postorder r @ [x]"
 
 text\<open>Binary Search Tree:\<close>
 fun bst_wrt :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a tree \<Rightarrow> bool" where
 "bst_wrt P \<langle>\<rangle> \<longleftrightarrow> True" |
 "bst_wrt P \<langle>l, a, r\<rangle> \<longleftrightarrow>
  bst_wrt P l \<and> bst_wrt P r \<and> (\<forall>x\<in>set_tree l. P x a) \<and> (\<forall>x\<in>set_tree r. P a x)"
 
 abbreviation bst :: "('a::linorder) tree \<Rightarrow> bool" where
 "bst \<equiv> bst_wrt (<)"
 
 fun (in linorder) heap :: "'a tree \<Rightarrow> bool" where
 "heap Leaf = True" |
 "heap (Node l m r) =
   (heap l \<and> heap r \<and> (\<forall>x \<in> set_tree l \<union> set_tree r. m \<le> x))"
 
 
 subsection \<open>\<^const>\<open>map_tree\<close>\<close>
 
 lemma eq_map_tree_Leaf[simp]: "map_tree f t = Leaf \<longleftrightarrow> t = Leaf"
 by (rule tree.map_disc_iff)
 
 lemma eq_Leaf_map_tree[simp]: "Leaf = map_tree f t \<longleftrightarrow> t = Leaf"
 by (cases t) auto
 
 
 subsection \<open>\<^const>\<open>size\<close>\<close>
 
 lemma size1_size: "size1 t = size t + 1"
 by (induction t) simp_all
 
 lemma size1_ge0[simp]: "0 < size1 t"
 by (simp add: size1_size)
 
 lemma eq_size_0[simp]: "size t = 0 \<longleftrightarrow> t = Leaf"
 by(cases t) auto
 
 lemma eq_0_size[simp]: "0 = size t \<longleftrightarrow> t = Leaf"
 by(cases t) auto
 
 lemma neq_Leaf_iff: "(t \<noteq> \<langle>\<rangle>) = (\<exists>l a r. t = \<langle>l, a, r\<rangle>)"
 by (cases t) auto
 
 lemma size_map_tree[simp]: "size (map_tree f t) = size t"
 by (induction t) auto
 
 lemma size1_map_tree[simp]: "size1 (map_tree f t) = size1 t"
 by (simp add: size1_size)
 
 
 subsection \<open>\<^const>\<open>set_tree\<close>\<close>
 
 lemma eq_set_tree_empty[simp]: "set_tree t = {} \<longleftrightarrow> t = Leaf"
 by (cases t) auto
 
 lemma eq_empty_set_tree[simp]: "{} = set_tree t \<longleftrightarrow> t = Leaf"
 by (cases t) auto
 
 lemma finite_set_tree[simp]: "finite(set_tree t)"
 by(induction t) auto
 
 
 subsection \<open>\<^const>\<open>subtrees\<close>\<close>
 
 lemma neq_subtrees_empty[simp]: "subtrees t \<noteq> {}"
 by (cases t)(auto)
 
 lemma neq_empty_subtrees[simp]: "{} \<noteq> subtrees t"
 by (cases t)(auto)
 
+lemma size_subtrees: "s \<in> subtrees t \<Longrightarrow> size s \<le> size t"
+by(induction t)(auto)
+
 lemma set_treeE: "a \<in> set_tree t \<Longrightarrow> \<exists>l r. \<langle>l, a, r\<rangle> \<in> subtrees t"
 by (induction t)(auto)
 
 lemma Node_notin_subtrees_if[simp]: "a \<notin> set_tree t \<Longrightarrow> Node l a r \<notin> subtrees t"
 by (induction t) auto
 
 lemma in_set_tree_if: "\<langle>l, a, r\<rangle> \<in> subtrees t \<Longrightarrow> a \<in> set_tree t"
 by (metis Node_notin_subtrees_if)
 
 
 subsection \<open>\<^const>\<open>height\<close> and \<^const>\<open>min_height\<close>\<close>
 
 lemma eq_height_0[simp]: "height t = 0 \<longleftrightarrow> t = Leaf"
 by(cases t) auto
 
 lemma eq_0_height[simp]: "0 = height t \<longleftrightarrow> t = Leaf"
 by(cases t) auto
 
 lemma height_map_tree[simp]: "height (map_tree f t) = height t"
 by (induction t) auto
 
 lemma height_le_size_tree: "height t \<le> size (t::'a tree)"
 by (induction t) auto
 
 lemma size1_height: "size1 t \<le> 2 ^ height (t::'a tree)"
 proof(induction t)
   case (Node l a r)
   show ?case
   proof (cases "height l \<le> height r")
     case True
     have "size1(Node l a r) = size1 l + size1 r" by simp
     also have "\<dots> \<le> 2 ^ height l + 2 ^ height r" using Node.IH by arith
     also have "\<dots> \<le> 2 ^ height r + 2 ^ height r" using True by simp
     also have "\<dots> = 2 ^ height (Node l a r)"
       using True by (auto simp: max_def mult_2)
     finally show ?thesis .
   next
     case False
     have "size1(Node l a r) = size1 l + size1 r" by simp
     also have "\<dots> \<le> 2 ^ height l + 2 ^ height r" using Node.IH by arith
     also have "\<dots> \<le> 2 ^ height l + 2 ^ height l" using False by simp
     finally show ?thesis using False by (auto simp: max_def mult_2)
   qed
 qed simp
 
 corollary size_height: "size t \<le> 2 ^ height (t::'a tree) - 1"
 using size1_height[of t, unfolded size1_size] by(arith)
 
 lemma height_subtrees: "s \<in> subtrees t \<Longrightarrow> height s \<le> height t"
 by (induction t) auto
 
 
 lemma min_height_le_height: "min_height t \<le> height t"
 by(induction t) auto
 
 lemma min_height_map_tree[simp]: "min_height (map_tree f t) = min_height t"
 by (induction t) auto
 
 lemma min_height_size1: "2 ^ min_height t \<le> size1 t"
 proof(induction t)
   case (Node l a r)
   have "(2::nat) ^ min_height (Node l a r) \<le> 2 ^ min_height l + 2 ^ min_height r"
     by (simp add: min_def)
   also have "\<dots> \<le> size1(Node l a r)" using Node.IH by simp
   finally show ?case .
 qed simp
 
 
 subsection \<open>\<^const>\<open>complete\<close>\<close>
 
 lemma complete_iff_height: "complete t \<longleftrightarrow> (min_height t = height t)"
 apply(induction t)
  apply simp
 apply (simp add: min_def max_def)
 by (metis le_antisym le_trans min_height_le_height)
 
 lemma size1_if_complete: "complete t \<Longrightarrow> size1 t = 2 ^ height t"
 by (induction t) auto
 
 lemma size_if_complete: "complete t \<Longrightarrow> size t = 2 ^ height t - 1"
 using size1_if_complete[simplified size1_size] by fastforce
 
 lemma size1_height_if_incomplete:
   "\<not> complete t \<Longrightarrow> size1 t < 2 ^ height t"
 proof(induction t)
   case Leaf thus ?case by simp
 next
   case (Node l x r)
   have 1: ?case if h: "height l < height r"
     using h size1_height[of l] size1_height[of r] power_strict_increasing[OF h, of "2::nat"]
     by(auto simp: max_def simp del: power_strict_increasing_iff)
   have 2: ?case if h: "height l > height r"
     using h size1_height[of l] size1_height[of r] power_strict_increasing[OF h, of "2::nat"]
     by(auto simp: max_def simp del: power_strict_increasing_iff)
   have 3: ?case if h: "height l = height r" and c: "\<not> complete l"
     using h size1_height[of r] Node.IH(1)[OF c] by(simp)
   have 4: ?case if h: "height l = height r" and c: "\<not> complete r"
     using h size1_height[of l] Node.IH(2)[OF c] by(simp)
   from 1 2 3 4 Node.prems show ?case apply (simp add: max_def) by linarith
 qed
 
 lemma complete_iff_min_height: "complete t \<longleftrightarrow> (height t = min_height t)"
 by(auto simp add: complete_iff_height)
 
 lemma min_height_size1_if_incomplete:
   "\<not> complete t \<Longrightarrow> 2 ^ min_height t < size1 t"
 proof(induction t)
   case Leaf thus ?case by simp
 next
   case (Node l x r)
   have 1: ?case if h: "min_height l < min_height r"
     using h min_height_size1[of l] min_height_size1[of r] power_strict_increasing[OF h, of "2::nat"]
     by(auto simp: max_def simp del: power_strict_increasing_iff)
   have 2: ?case if h: "min_height l > min_height r"
     using h min_height_size1[of l] min_height_size1[of r] power_strict_increasing[OF h, of "2::nat"]
     by(auto simp: max_def simp del: power_strict_increasing_iff)
   have 3: ?case if h: "min_height l = min_height r" and c: "\<not> complete l"
     using h min_height_size1[of r] Node.IH(1)[OF c] by(simp add: complete_iff_min_height)
   have 4: ?case if h: "min_height l = min_height r" and c: "\<not> complete r"
     using h min_height_size1[of l] Node.IH(2)[OF c] by(simp add: complete_iff_min_height)
   from 1 2 3 4 Node.prems show ?case
     by (fastforce simp: complete_iff_min_height[THEN iffD1])
 qed
 
 lemma complete_if_size1_height: "size1 t = 2 ^ height t \<Longrightarrow> complete t"
 using  size1_height_if_incomplete by fastforce
 
 lemma complete_if_size1_min_height: "size1 t = 2 ^ min_height t \<Longrightarrow> complete t"
 using min_height_size1_if_incomplete by fastforce
 
 lemma complete_iff_size1: "complete t \<longleftrightarrow> size1 t = 2 ^ height t"
 using complete_if_size1_height size1_if_complete by blast
 
 
 subsection \<open>\<^const>\<open>balanced\<close>\<close>
 
 lemma balanced_subtreeL: "balanced (Node l x r) \<Longrightarrow> balanced l"
 by(simp add: balanced_def)
 
 lemma balanced_subtreeR: "balanced (Node l x r) \<Longrightarrow> balanced r"
 by(simp add: balanced_def)
 
 lemma balanced_subtrees: "\<lbrakk> balanced t; s \<in> subtrees t \<rbrakk> \<Longrightarrow> balanced s"
 using [[simp_depth_limit=1]]
 by(induction t arbitrary: s)
   (auto simp add: balanced_subtreeL balanced_subtreeR)
 
 text\<open>Balanced trees have optimal height:\<close>
 
 lemma balanced_optimal:
 fixes t :: "'a tree" and t' :: "'b tree"
 assumes "balanced t" "size t \<le> size t'" shows "height t \<le> height t'"
 proof (cases "complete t")
   case True
   have "(2::nat) ^ height t \<le> 2 ^ height t'"
   proof -
     have "2 ^ height t = size1 t"
       using True by (simp add: size1_if_complete)
     also have "\<dots> \<le> size1 t'" using assms(2) by(simp add: size1_size)
     also have "\<dots> \<le> 2 ^ height t'" by (rule size1_height)
     finally show ?thesis .
   qed
   thus ?thesis by (simp)
 next
   case False
   have "(2::nat) ^ min_height t < 2 ^ height t'"
   proof -
     have "(2::nat) ^ min_height t < size1 t"
       by(rule min_height_size1_if_incomplete[OF False])
     also have "\<dots> \<le> size1 t'" using assms(2) by (simp add: size1_size)
     also have "\<dots> \<le> 2 ^ height t'"  by(rule size1_height)
     finally have "(2::nat) ^ min_height t < (2::nat) ^ height t'" .
     thus ?thesis .
   qed
   hence *: "min_height t < height t'" by simp
   have "min_height t + 1 = height t"
     using min_height_le_height[of t] assms(1) False
     by (simp add: complete_iff_height balanced_def)
   with * show ?thesis by arith
 qed
 
 
 subsection \<open>\<^const>\<open>wbalanced\<close>\<close>
 
 lemma wbalanced_subtrees: "\<lbrakk> wbalanced t; s \<in> subtrees t \<rbrakk> \<Longrightarrow> wbalanced s"
 using [[simp_depth_limit=1]] by(induction t arbitrary: s) auto
 
 
 subsection \<open>\<^const>\<open>ipl\<close>\<close>
 
 text \<open>The internal path length of a tree:\<close>
 
 lemma ipl_if_complete_int:
   "complete t \<Longrightarrow> int(ipl t) = (int(height t) - 2) * 2^(height t) + 2"
 apply(induction t)
  apply simp
 apply simp
 apply (simp add: algebra_simps size_if_complete of_nat_diff)
 done
 
 
 subsection "List of entries"
 
 lemma eq_inorder_Nil[simp]: "inorder t = [] \<longleftrightarrow> t = Leaf"
 by (cases t) auto
 
 lemma eq_Nil_inorder[simp]: "[] = inorder t \<longleftrightarrow> t = Leaf"
 by (cases t) auto
 
 lemma set_inorder[simp]: "set (inorder t) = set_tree t"
 by (induction t) auto
 
 lemma set_preorder[simp]: "set (preorder t) = set_tree t"
 by (induction t) auto
 
 lemma set_postorder[simp]: "set (postorder t) = set_tree t"
 by (induction t) auto
 
 lemma length_preorder[simp]: "length (preorder t) = size t"
 by (induction t) auto
 
 lemma length_inorder[simp]: "length (inorder t) = size t"
 by (induction t) auto
 
 lemma length_postorder[simp]: "length (postorder t) = size t"
 by (induction t) auto
 
 lemma preorder_map: "preorder (map_tree f t) = map f (preorder t)"
 by (induction t) auto
 
 lemma inorder_map: "inorder (map_tree f t) = map f (inorder t)"
 by (induction t) auto
 
 lemma postorder_map: "postorder (map_tree f t) = map f (postorder t)"
 by (induction t) auto
 
 lemma inorder2_inorder: "inorder2 t xs = inorder t @ xs"
 by (induction t arbitrary: xs) auto
 
 
 subsection \<open>Binary Search Tree\<close>
 
 lemma bst_wrt_mono: "(\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> bst_wrt P t \<Longrightarrow> bst_wrt Q t"
 by (induction t) (auto)
 
 lemma bst_wrt_le_if_bst: "bst t \<Longrightarrow> bst_wrt (\<le>) t"
 using bst_wrt_mono less_imp_le by blast
 
 lemma bst_wrt_le_iff_sorted: "bst_wrt (\<le>) t \<longleftrightarrow> sorted (inorder t)"
 apply (induction t)
  apply(simp)
 by (fastforce simp: sorted_append intro: less_imp_le less_trans)
 
 lemma bst_iff_sorted_wrt_less: "bst t \<longleftrightarrow> sorted_wrt (<) (inorder t)"
 apply (induction t)
  apply simp
 apply (fastforce simp: sorted_wrt_append)
 done
 
 
 subsection \<open>\<^const>\<open>heap\<close>\<close>
 
 
 subsection \<open>\<^const>\<open>mirror\<close>\<close>
 
 lemma mirror_Leaf[simp]: "mirror t = \<langle>\<rangle> \<longleftrightarrow> t = \<langle>\<rangle>"
 by (induction t) simp_all
 
 lemma Leaf_mirror[simp]: "\<langle>\<rangle> = mirror t \<longleftrightarrow> t = \<langle>\<rangle>"
 using mirror_Leaf by fastforce
 
 lemma size_mirror[simp]: "size(mirror t) = size t"
 by (induction t) simp_all
 
 lemma size1_mirror[simp]: "size1(mirror t) = size1 t"
 by (simp add: size1_size)
 
 lemma height_mirror[simp]: "height(mirror t) = height t"
 by (induction t) simp_all
 
 lemma min_height_mirror [simp]: "min_height (mirror t) = min_height t"
 by (induction t) simp_all  
 
 lemma ipl_mirror [simp]: "ipl (mirror t) = ipl t"
 by (induction t) simp_all
 
 lemma inorder_mirror: "inorder(mirror t) = rev(inorder t)"
 by (induction t) simp_all
 
 lemma map_mirror: "map_tree f (mirror t) = mirror (map_tree f t)"
 by (induction t) simp_all
 
 lemma mirror_mirror[simp]: "mirror(mirror t) = t"
 by (induction t) simp_all
 
 end