src/HOL/Library/Tree.thy
author nipkow
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(* Author: Tobias Nipkow *)
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(* Todo: minimal ipl of balanced trees *)
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section \<open>Binary Tree\<close>
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theory Tree
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imports Main
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begin
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datatype 'a tree =
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  Leaf ("\<langle>\<rangle>") |
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  Node "'a tree" (root_val: 'a) "'a tree" ("(1\<langle>_,/ _,/ _\<rangle>)")
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datatype_compat tree
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text\<open>Can be seen as counting the number of leaves rather than nodes:\<close>
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definition size1 :: "'a tree \<Rightarrow> nat" where
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"size1 t = size t + 1"
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fun subtrees :: "'a tree \<Rightarrow> 'a tree set" where
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"subtrees \<langle>\<rangle> = {\<langle>\<rangle>}" |
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"subtrees (\<langle>l, a, r\<rangle>) = insert \<langle>l, a, r\<rangle> (subtrees l \<union> subtrees r)"
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fun mirror :: "'a tree \<Rightarrow> 'a tree" where
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"mirror \<langle>\<rangle> = Leaf" |
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"mirror \<langle>l,x,r\<rangle> = \<langle>mirror r, x, mirror l\<rangle>"
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class height = fixes height :: "'a \<Rightarrow> nat"
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instantiation tree :: (type)height
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begin
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fun height_tree :: "'a tree => nat" where
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"height Leaf = 0" |
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"height (Node t1 a t2) = max (height t1) (height t2) + 1"
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instance ..
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end
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fun min_height :: "'a tree \<Rightarrow> nat" where
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"min_height Leaf = 0" |
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"min_height (Node l _ r) = min (min_height l) (min_height r) + 1"
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fun complete :: "'a tree \<Rightarrow> bool" where
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"complete Leaf = True" |
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"complete (Node l x r) = (complete l \<and> complete r \<and> height l = height r)"
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definition balanced :: "'a tree \<Rightarrow> bool" where
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"balanced t = (height t - min_height t \<le> 1)"
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text \<open>Weight balanced:\<close>
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fun wbalanced :: "'a tree \<Rightarrow> bool" where
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"wbalanced Leaf = True" |
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"wbalanced (Node l x r) = (abs(int(size l) - int(size r)) \<le> 1 \<and> wbalanced l \<and> wbalanced r)"
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text \<open>Internal path length:\<close>
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fun ipl :: "'a tree \<Rightarrow> nat" where
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"ipl Leaf = 0 " |
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"ipl (Node l _ r) = ipl l + size l + ipl r + size r"
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fun preorder :: "'a tree \<Rightarrow> 'a list" where
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"preorder \<langle>\<rangle> = []" |
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"preorder \<langle>l, x, r\<rangle> = x # preorder l @ preorder r"
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fun inorder :: "'a tree \<Rightarrow> 'a list" where
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"inorder \<langle>\<rangle> = []" |
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"inorder \<langle>l, x, r\<rangle> = inorder l @ [x] @ inorder r"
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text\<open>A linear version avoiding append:\<close>
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fun inorder2 :: "'a tree \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"inorder2 \<langle>\<rangle> xs = xs" |
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"inorder2 \<langle>l, x, r\<rangle> xs = inorder2 l (x # inorder2 r xs)"
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text\<open>Binary Search Tree:\<close>
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fun (in linorder) bst :: "'a tree \<Rightarrow> bool" where
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"bst \<langle>\<rangle> \<longleftrightarrow> True" |
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"bst \<langle>l, a, r\<rangle> \<longleftrightarrow> bst l \<and> bst r \<and> (\<forall>x\<in>set_tree l. x < a) \<and> (\<forall>x\<in>set_tree r. a < x)"
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text\<open>Binary Search Tree with duplicates:\<close>
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fun (in linorder) bst_eq :: "'a tree \<Rightarrow> bool" where
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"bst_eq \<langle>\<rangle> \<longleftrightarrow> True" |
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"bst_eq \<langle>l,a,r\<rangle> \<longleftrightarrow>
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 bst_eq l \<and> bst_eq r \<and> (\<forall>x\<in>set_tree l. x \<le> a) \<and> (\<forall>x\<in>set_tree r. a \<le> x)"
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fun (in linorder) heap :: "'a tree \<Rightarrow> bool" where
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"heap Leaf = True" |
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"heap (Node l m r) =
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  (heap l \<and> heap r \<and> (\<forall>x \<in> set_tree l \<union> set_tree r. m \<le> x))"
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subsection \<open>@{const size}\<close>
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lemma size1_simps[simp]:
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  "size1 \<langle>\<rangle> = 1"
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  "size1 \<langle>l, x, r\<rangle> = size1 l + size1 r"
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by (simp_all add: size1_def)
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lemma size1_ge0[simp]: "0 < size1 t"
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by (simp add: size1_def)
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lemma size_0_iff_Leaf: "size t = 0 \<longleftrightarrow> t = Leaf"
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by(cases t) auto
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lemma neq_Leaf_iff: "(t \<noteq> \<langle>\<rangle>) = (\<exists>l a r. t = \<langle>l, a, r\<rangle>)"
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by (cases t) auto
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lemma finite_set_tree[simp]: "finite(set_tree t)"
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by(induction t) auto
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lemma size_map_tree[simp]: "size (map_tree f t) = size t"
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by (induction t) auto
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lemma size1_map_tree[simp]: "size1 (map_tree f t) = size1 t"
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by (simp add: size1_def)
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subsection \<open>@{const subtrees}\<close>
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lemma set_treeE: "a \<in> set_tree t \<Longrightarrow> \<exists>l r. \<langle>l, a, r\<rangle> \<in> subtrees t"
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by (induction t)(auto)
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lemma Node_notin_subtrees_if[simp]: "a \<notin> set_tree t \<Longrightarrow> Node l a r \<notin> subtrees t"
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by (induction t) auto
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lemma in_set_tree_if: "\<langle>l, a, r\<rangle> \<in> subtrees t \<Longrightarrow> a \<in> set_tree t"
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by (metis Node_notin_subtrees_if)
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subsection \<open>@{const height} and @{const min_height}\<close>
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lemma height_0_iff_Leaf: "height t = 0 \<longleftrightarrow> t = Leaf"
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by(cases t) auto
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lemma height_map_tree[simp]: "height (map_tree f t) = height t"
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by (induction t) auto
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lemma height_le_size_tree: "height t \<le> size (t::'a tree)"
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by (induction t) auto
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lemma size1_height: "size1 t \<le> 2 ^ height (t::'a tree)"
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proof(induction t)
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  case (Node l a r)
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  show ?case
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  proof (cases "height l \<le> height r")
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    case True
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    have "size1(Node l a r) = size1 l + size1 r" by simp
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    also have "\<dots> \<le> 2 ^ height l + 2 ^ height r" using Node.IH by arith
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    also have "\<dots> \<le> 2 ^ height r + 2 ^ height r" using True by simp
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    also have "\<dots> = 2 ^ height (Node l a r)"
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      using True by (auto simp: max_def mult_2)
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    finally show ?thesis .
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  next
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    case False
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    have "size1(Node l a r) = size1 l + size1 r" by simp
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    also have "\<dots> \<le> 2 ^ height l + 2 ^ height r" using Node.IH by arith
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    also have "\<dots> \<le> 2 ^ height l + 2 ^ height l" using False by simp
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    finally show ?thesis using False by (auto simp: max_def mult_2)
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  qed
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qed simp
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corollary size_height: "size t \<le> 2 ^ height (t::'a tree) - 1"
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using size1_height[of t, unfolded size1_def] by(arith)
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lemma height_subtrees: "s \<in> subtrees t \<Longrightarrow> height s \<le> height t"
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by (induction t) auto
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   167
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   168
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   169
lemma min_height_le_height: "min_height t \<le> height t"
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   170
by(induction t) auto
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   171
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   172
lemma min_height_map_tree[simp]: "min_height (map_tree f t) = min_height t"
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   173
by (induction t) auto
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diff changeset
   174
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   175
lemma min_height_size1: "2 ^ min_height t \<le> size1 t"
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   176
proof(induction t)
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   177
  case (Node l a r)
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   178
  have "(2::nat) ^ min_height (Node l a r) \<le> 2 ^ min_height l + 2 ^ min_height r"
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   179
    by (simp add: min_def)
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   180
  also have "\<dots> \<le> size1(Node l a r)" using Node.IH by simp
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   181
  finally show ?case .
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   182
qed simp
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   183
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   184
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   185
subsection \<open>@{const complete}\<close>
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   186
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lemma complete_iff_height: "complete t \<longleftrightarrow> (min_height t = height t)"
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   188
apply(induction t)
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   189
 apply simp
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   190
apply (simp add: min_def max_def)
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   191
by (metis le_antisym le_trans min_height_le_height)
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   192
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   193
lemma size1_if_complete: "complete t \<Longrightarrow> size1 t = 2 ^ height t"
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   194
by (induction t) auto
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   195
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   196
lemma size_if_complete: "complete t \<Longrightarrow> size t = 2 ^ height t - 1"
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   197
using size1_if_complete[simplified size1_def] by fastforce
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   198
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   199
lemma complete_if_size1_height: "size1 t = 2 ^ height t \<Longrightarrow> complete t"
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   200
proof (induct "height t" arbitrary: t)
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   201
  case 0 thus ?case by (simp add: height_0_iff_Leaf)
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   202
next
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   203
  case (Suc h)
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   204
  hence "t \<noteq> Leaf" by auto
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   205
  then obtain l a r where [simp]: "t = Node l a r"
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   206
    by (auto simp: neq_Leaf_iff)
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   207
  have 1: "height l \<le> h" and 2: "height r \<le> h" using Suc(2) by(auto)
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   208
  have 3: "\<not> height l < h"
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   209
  proof
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   210
    assume 0: "height l < h"
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   211
    have "size1 t = size1 l + size1 r" by simp
64918
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   212
    also have "\<dots> \<le> 2 ^ height l + 2 ^ height r"
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   213
      using size1_height[of l] size1_height[of r] by arith
nipkow
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   214
    also have " \<dots> < 2 ^ h + 2 ^ height r" using 0 by (simp)
nipkow
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diff changeset
   215
    also have " \<dots> \<le> 2 ^ h + 2 ^ h" using 2 by (simp)
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   216
    also have "\<dots> = 2 ^ (Suc h)" by (simp)
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   217
    also have "\<dots> = size1 t" using Suc(2,3) by simp
64918
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   218
    finally have "size1 t < size1 t" .
nipkow
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   219
    thus False by (simp)
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   220
  qed
64918
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   221
  have 4: "\<not> height r < h"
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   222
  proof
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   223
    assume 0: "height r < h"
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   224
    have "size1 t = size1 l + size1 r" by simp
64918
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   225
    also have "\<dots> \<le> 2 ^ height l + 2 ^ height r"
nipkow
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   226
      using size1_height[of l] size1_height[of r] by arith
nipkow
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   227
    also have " \<dots> < 2 ^ height l + 2 ^ h" using 0 by (simp)
nipkow
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diff changeset
   228
    also have " \<dots> \<le> 2 ^ h + 2 ^ h" using 1 by (simp)
nipkow
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   229
    also have "\<dots> = 2 ^ (Suc h)" by (simp)
64533
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   230
    also have "\<dots> = size1 t" using Suc(2,3) by simp
64918
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   231
    finally have "size1 t < size1 t" .
nipkow
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diff changeset
   232
    thus False by (simp)
63770
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   233
  qed
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diff changeset
   234
  from 1 2 3 4 have *: "height l = h" "height r = h" by linarith+
64533
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   235
  hence "size1 l = 2 ^ height l" "size1 r = 2 ^ height r"
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   236
    using Suc(3) size1_height[of l] size1_height[of r] by (auto)
63770
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   237
  with * Suc(1) show ?case by simp
a67397b13eb5 added lemmas
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   238
qed
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diff changeset
   239
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   240
text\<open>The following proof involves \<open>\<ge>\<close>/\<open>>\<close> chains rather than the standard
172f3a047f4a more lemmas, tuned proofs
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   241
\<open>\<le>\<close>/\<open><\<close> chains. To chain the elements together the transitivity rules \<open>xtrans\<close>
172f3a047f4a more lemmas, tuned proofs
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   242
are used.\<close>
63755
182c111190e5 Renamed balanced to complete; added balanced; more about both
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diff changeset
   243
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   244
lemma complete_if_size1_min_height: "size1 t = 2 ^ min_height t \<Longrightarrow> complete t"
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   245
proof (induct "min_height t" arbitrary: t)
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   246
  case 0 thus ?case by (simp add: size_0_iff_Leaf size1_def)
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   247
next
64533
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   248
  case (Suc h)
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   249
  hence "t \<noteq> Leaf" by auto
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   250
  then obtain l a r where [simp]: "t = Node l a r"
172f3a047f4a more lemmas, tuned proofs
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   251
    by (auto simp: neq_Leaf_iff)
172f3a047f4a more lemmas, tuned proofs
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   252
  have 1: "h \<le> min_height l" and 2: "h \<le> min_height r" using Suc(2) by(auto)
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   253
  have 3: "\<not> h < min_height l"
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   254
  proof
172f3a047f4a more lemmas, tuned proofs
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   255
    assume 0: "h < min_height l"
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   256
    have "size1 t = size1 l + size1 r" by simp
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   257
    also note min_height_size1[of l]
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   258
    also(xtrans) note min_height_size1[of r]
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   259
    also(xtrans) have "(2::nat) ^ min_height l > 2 ^ h"
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   260
        using 0 by (simp add: diff_less_mono)
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   261
    also(xtrans) have "(2::nat) ^ min_height r \<ge> 2 ^ h" using 2 by simp
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   262
    also(xtrans) have "(2::nat) ^ h + 2 ^ h = 2 ^ (Suc h)" by (simp)
172f3a047f4a more lemmas, tuned proofs
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parents: 64414
diff changeset
   263
    also have "\<dots> = size1 t" using Suc(2,3) by simp
172f3a047f4a more lemmas, tuned proofs
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parents: 64414
diff changeset
   264
    finally show False by (simp add: diff_le_mono)
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182c111190e5 Renamed balanced to complete; added balanced; more about both
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   265
  qed
64533
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diff changeset
   266
  have 4: "\<not> h < min_height r"
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diff changeset
   267
  proof
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   268
    assume 0: "h < min_height r"
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   269
    have "size1 t = size1 l + size1 r" by simp
172f3a047f4a more lemmas, tuned proofs
nipkow
parents: 64414
diff changeset
   270
    also note min_height_size1[of l]
172f3a047f4a more lemmas, tuned proofs
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parents: 64414
diff changeset
   271
    also(xtrans) note min_height_size1[of r]
172f3a047f4a more lemmas, tuned proofs
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parents: 64414
diff changeset
   272
    also(xtrans) have "(2::nat) ^ min_height r > 2 ^ h"
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   273
        using 0 by (simp add: diff_less_mono)
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   274
    also(xtrans) have "(2::nat) ^ min_height l \<ge> 2 ^ h" using 1 by simp
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   275
    also(xtrans) have "(2::nat) ^ h + 2 ^ h = 2 ^ (Suc h)" by (simp)
172f3a047f4a more lemmas, tuned proofs
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parents: 64414
diff changeset
   276
    also have "\<dots> = size1 t" using Suc(2,3) by simp
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   277
    finally show False by (simp add: diff_le_mono)
172f3a047f4a more lemmas, tuned proofs
nipkow
parents: 64414
diff changeset
   278
  qed
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   279
  from 1 2 3 4 have *: "min_height l = h" "min_height r = h" by linarith+
172f3a047f4a more lemmas, tuned proofs
nipkow
parents: 64414
diff changeset
   280
  hence "size1 l = 2 ^ min_height l" "size1 r = 2 ^ min_height r"
172f3a047f4a more lemmas, tuned proofs
nipkow
parents: 64414
diff changeset
   281
    using Suc(3) min_height_size1[of l] min_height_size1[of r] by (auto)
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   282
  with * Suc(1) show ?case
172f3a047f4a more lemmas, tuned proofs
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parents: 64414
diff changeset
   283
    by (simp add: complete_iff_height)
63755
182c111190e5 Renamed balanced to complete; added balanced; more about both
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diff changeset
   284
qed
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   285
64533
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diff changeset
   286
lemma complete_iff_size1: "complete t \<longleftrightarrow> size1 t = 2 ^ height t"
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   287
using complete_if_size1_height size1_if_complete by blast
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   288
172f3a047f4a more lemmas, tuned proofs
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   289
text\<open>Better bounds for incomplete trees:\<close>
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   290
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   291
lemma size1_height_if_incomplete:
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   292
  "\<not> complete t \<Longrightarrow> size1 t < 2 ^ height t"
172f3a047f4a more lemmas, tuned proofs
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parents: 64414
diff changeset
   293
by (meson antisym_conv complete_iff_size1 not_le size1_height)
172f3a047f4a more lemmas, tuned proofs
nipkow
parents: 64414
diff changeset
   294
172f3a047f4a more lemmas, tuned proofs
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diff changeset
   295
lemma min_height_size1_if_incomplete:
172f3a047f4a more lemmas, tuned proofs
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parents: 64414
diff changeset
   296
  "\<not> complete t \<Longrightarrow> 2 ^ min_height t < size1 t"
172f3a047f4a more lemmas, tuned proofs
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parents: 64414
diff changeset
   297
by (metis complete_if_size1_min_height le_less min_height_size1)
172f3a047f4a more lemmas, tuned proofs
nipkow
parents: 64414
diff changeset
   298
63755
182c111190e5 Renamed balanced to complete; added balanced; more about both
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diff changeset
   299
63861
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   300
subsection \<open>@{const balanced}\<close>
90360390a916 reorganization, more funs and lemmas
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diff changeset
   301
90360390a916 reorganization, more funs and lemmas
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   302
lemma balanced_subtreeL: "balanced (Node l x r) \<Longrightarrow> balanced l"
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   303
by(simp add: balanced_def)
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182c111190e5 Renamed balanced to complete; added balanced; more about both
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parents: 63665
diff changeset
   304
63861
90360390a916 reorganization, more funs and lemmas
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diff changeset
   305
lemma balanced_subtreeR: "balanced (Node l x r) \<Longrightarrow> balanced r"
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diff changeset
   306
by(simp add: balanced_def)
90360390a916 reorganization, more funs and lemmas
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diff changeset
   307
90360390a916 reorganization, more funs and lemmas
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diff changeset
   308
lemma balanced_subtrees: "\<lbrakk> balanced t; s \<in> subtrees t \<rbrakk> \<Longrightarrow> balanced s"
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parents: 63829
diff changeset
   309
using [[simp_depth_limit=1]]
90360390a916 reorganization, more funs and lemmas
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diff changeset
   310
by(induction t arbitrary: s)
90360390a916 reorganization, more funs and lemmas
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diff changeset
   311
  (auto simp add: balanced_subtreeL balanced_subtreeR)
63755
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   312
182c111190e5 Renamed balanced to complete; added balanced; more about both
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diff changeset
   313
text\<open>Balanced trees have optimal height:\<close>
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   314
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   315
lemma balanced_optimal:
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   316
fixes t :: "'a tree" and t' :: "'b tree"
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   317
assumes "balanced t" "size t \<le> size t'" shows "height t \<le> height t'"
182c111190e5 Renamed balanced to complete; added balanced; more about both
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parents: 63665
diff changeset
   318
proof (cases "complete t")
182c111190e5 Renamed balanced to complete; added balanced; more about both
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parents: 63665
diff changeset
   319
  case True
64924
nipkow
parents: 64923
diff changeset
   320
  have "(2::nat) ^ height t \<le> 2 ^ height t'"
63755
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   321
  proof -
64924
nipkow
parents: 64923
diff changeset
   322
    have "2 ^ height t = size1 t"
nipkow
parents: 64923
diff changeset
   323
      using True by (simp add: complete_iff_height size1_if_complete)
nipkow
parents: 64923
diff changeset
   324
    also have "\<dots> \<le> size1 t'" using assms(2) by(simp add: size1_def)
nipkow
parents: 64923
diff changeset
   325
    also have "\<dots> \<le> 2 ^ height t'" by (rule size1_height)
63755
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   326
    finally show ?thesis .
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   327
  qed
64924
nipkow
parents: 64923
diff changeset
   328
  thus ?thesis by (simp)
63755
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   329
next
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   330
  case False
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   331
  have "(2::nat) ^ min_height t < 2 ^ height t'"
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   332
  proof -
64533
172f3a047f4a more lemmas, tuned proofs
nipkow
parents: 64414
diff changeset
   333
    have "(2::nat) ^ min_height t < size1 t"
172f3a047f4a more lemmas, tuned proofs
nipkow
parents: 64414
diff changeset
   334
      by(rule min_height_size1_if_incomplete[OF False])
64918
nipkow
parents: 64887
diff changeset
   335
    also have "\<dots> \<le> size1 t'" using assms(2) by (simp add: size1_def)
nipkow
parents: 64887
diff changeset
   336
    also have "\<dots> \<le> 2 ^ height t'"  by(rule size1_height)
nipkow
parents: 64887
diff changeset
   337
    finally have "(2::nat) ^ min_height t < (2::nat) ^ height t'" .
64924
nipkow
parents: 64923
diff changeset
   338
    thus ?thesis .
63755
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   339
  qed
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   340
  hence *: "min_height t < height t'" by simp
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   341
  have "min_height t + 1 = height t"
64540
f1f4ba6d02c9 spelling
nipkow
parents: 64533
diff changeset
   342
    using min_height_le_height[of t] assms(1) False
63829
6a05c8cbf7de More on balancing; renamed theory to Balance
nipkow
parents: 63770
diff changeset
   343
    by (simp add: complete_iff_height balanced_def)
63755
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   344
  with * show ?thesis by arith
182c111190e5 Renamed balanced to complete; added balanced; more about both
nipkow
parents: 63665
diff changeset
   345
qed
63036
1ba3aacfa4d3 added "balanced" predicate
nipkow
parents: 62650
diff changeset
   346
1ba3aacfa4d3 added "balanced" predicate
nipkow
parents: 62650
diff changeset
   347
63861
90360390a916 reorganization, more funs and lemmas
nipkow
parents: 63829
diff changeset
   348
subsection \<open>@{const wbalanced}\<close>
90360390a916 reorganization, more funs and lemmas
nipkow
parents: 63829
diff changeset
   349
90360390a916 reorganization, more funs and lemmas
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parents: 63829
diff changeset
   350
lemma wbalanced_subtrees: "\<lbrakk> wbalanced t; s \<in> subtrees t \<rbrakk> \<Longrightarrow> wbalanced s"
90360390a916 reorganization, more funs and lemmas
nipkow
parents: 63829
diff changeset
   351
using [[simp_depth_limit=1]] by(induction t arbitrary: s) auto
90360390a916 reorganization, more funs and lemmas
nipkow
parents: 63829
diff changeset
   352
90360390a916 reorganization, more funs and lemmas
nipkow
parents: 63829
diff changeset
   353
64887
266fb24c80bd tuned/minimized
nipkow
parents: 64771
diff changeset
   354
subsection \<open>@{const ipl}\<close>
63413
9fe2d9dc095e added path_len
nipkow
parents: 63036
diff changeset
   355
9fe2d9dc095e added path_len
nipkow
parents: 63036
diff changeset
   356
text \<open>The internal path length of a tree:\<close>
9fe2d9dc095e added path_len
nipkow
parents: 63036
diff changeset
   357
64923
7c340dcbc323 int version slicker
nipkow
parents: 64922
diff changeset
   358
lemma ipl_if_complete_int:
7c340dcbc323 int version slicker
nipkow
parents: 64922
diff changeset
   359
  "complete t \<Longrightarrow> int(ipl t) = (int(height t) - 2) * 2^(height t) + 2"
7c340dcbc323 int version slicker
nipkow
parents: 64922
diff changeset
   360
apply(induction t)
7c340dcbc323 int version slicker
nipkow
parents: 64922
diff changeset
   361
 apply simp
7c340dcbc323 int version slicker
nipkow
parents: 64922
diff changeset
   362
apply simp
7c340dcbc323 int version slicker
nipkow
parents: 64922
diff changeset
   363
apply (simp add: algebra_simps size_if_complete of_nat_diff)
7c340dcbc323 int version slicker
nipkow
parents: 64922
diff changeset
   364
done
63413
9fe2d9dc095e added path_len
nipkow
parents: 63036
diff changeset
   365
9fe2d9dc095e added path_len
nipkow
parents: 63036
diff changeset
   366
59776
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   367
subsection "List of entries"
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   368
57449
f81da03b9ebd Library/Tree: use datatype_new, bst is an inductive predicate
hoelzl
parents: 57250
diff changeset
   369
lemma set_inorder[simp]: "set (inorder t) = set_tree t"
58424
cbbba613b6ab added nice standard syntax
nipkow
parents: 58310
diff changeset
   370
by (induction t) auto
57250
cddaf5b93728 new theory of binary trees
nipkow
parents:
diff changeset
   371
59776
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   372
lemma set_preorder[simp]: "set (preorder t) = set_tree t"
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   373
by (induction t) auto
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   374
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   375
lemma length_preorder[simp]: "length (preorder t) = size t"
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   376
by (induction t) auto
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   377
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   378
lemma length_inorder[simp]: "length (inorder t) = size t"
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   379
by (induction t) auto
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   380
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   381
lemma preorder_map: "preorder (map_tree f t) = map f (preorder t)"
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   382
by (induction t) auto
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   383
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   384
lemma inorder_map: "inorder (map_tree f t) = map f (inorder t)"
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   385
by (induction t) auto
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   386
63765
e60020520b15 added inorder2
nipkow
parents: 63755
diff changeset
   387
lemma inorder2_inorder: "inorder2 t xs = inorder t @ xs"
e60020520b15 added inorder2
nipkow
parents: 63755
diff changeset
   388
by (induction t arbitrary: xs) auto
e60020520b15 added inorder2
nipkow
parents: 63755
diff changeset
   389
57687
cca7e8788481 added more functions and lemmas
nipkow
parents: 57569
diff changeset
   390
63861
90360390a916 reorganization, more funs and lemmas
nipkow
parents: 63829
diff changeset
   391
subsection \<open>Binary Search Tree\<close>
59561
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   392
59928
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   393
lemma (in linorder) bst_eq_if_bst: "bst t \<Longrightarrow> bst_eq t"
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   394
by (induction t) (auto)
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   395
59561
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   396
lemma (in linorder) bst_eq_imp_sorted: "bst_eq t \<Longrightarrow> sorted (inorder t)"
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   397
apply (induction t)
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   398
 apply(simp)
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   399
by (fastforce simp: sorted_append sorted_Cons intro: less_imp_le less_trans)
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   400
59928
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   401
lemma (in linorder) distinct_preorder_if_bst: "bst t \<Longrightarrow> distinct (preorder t)"
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   402
apply (induction t)
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   403
 apply simp
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   404
apply(fastforce elim: order.asym)
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   405
done
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   406
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   407
lemma (in linorder) distinct_inorder_if_bst: "bst t \<Longrightarrow> distinct (inorder t)"
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   408
apply (induction t)
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   409
 apply simp
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   410
apply(fastforce elim: order.asym)
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   411
done
b9b7f913a19a new theory Library/Tree_Multiset.thy
nipkow
parents: 59776
diff changeset
   412
59776
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   413
63861
90360390a916 reorganization, more funs and lemmas
nipkow
parents: 63829
diff changeset
   414
subsection \<open>@{const heap}\<close>
60505
9e6584184315 added funs and lemmas
nipkow
parents: 59928
diff changeset
   415
9e6584184315 added funs and lemmas
nipkow
parents: 59928
diff changeset
   416
63861
90360390a916 reorganization, more funs and lemmas
nipkow
parents: 63829
diff changeset
   417
subsection \<open>@{const mirror}\<close>
59561
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   418
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   419
lemma mirror_Leaf[simp]: "mirror t = \<langle>\<rangle> \<longleftrightarrow> t = \<langle>\<rangle>"
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   420
by (induction t) simp_all
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   421
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   422
lemma size_mirror[simp]: "size(mirror t) = size t"
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   423
by (induction t) simp_all
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   424
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   425
lemma size1_mirror[simp]: "size1(mirror t) = size1 t"
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   426
by (simp add: size1_def)
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   427
60808
fd26519b1a6a depth -> height; removed del_rightmost (too specifi)
nipkow
parents: 60507
diff changeset
   428
lemma height_mirror[simp]: "height(mirror t) = height t"
59776
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   429
by (induction t) simp_all
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   430
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   431
lemma inorder_mirror: "inorder(mirror t) = rev(inorder t)"
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   432
by (induction t) simp_all
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   433
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   434
lemma map_mirror: "map_tree f (mirror t) = mirror (map_tree f t)"
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   435
by (induction t) simp_all
f54af3307334 added funs and lemmas
nipkow
parents: 59561
diff changeset
   436
59561
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   437
lemma mirror_mirror[simp]: "mirror(mirror t) = t"
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   438
by (induction t) simp_all
1a84beaa239b added new tree material
nipkow
parents: 58881
diff changeset
   439
57250
cddaf5b93728 new theory of binary trees
nipkow
parents:
diff changeset
   440
end