author | nipkow |
Thu, 04 Oct 2018 10:35:29 +0200 | |
changeset 69117 | 3d3e87835ae8 |
parent 69115 | 919a1b23c192 |
child 69218 | 59aefb3b9e95 |
permissions | -rw-r--r-- |
57250 | 1 |
(* 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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||
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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>Counting the number of leaves rather than nodes:\<close> |
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fun size1 :: "'a tree \<Rightarrow> nat" where |
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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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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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||
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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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||
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class height = fixes height :: "'a \<Rightarrow> nat" |
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||
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instantiation tree :: (type)height |
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begin |
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||
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fun height_tree :: "'a tree => nat" where |
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"height Leaf = 0" | |
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"height (Node l a r) = max (height l) (height r) + 1" |
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instance .. |
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||
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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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||
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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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||
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fun postorder :: "'a tree \<Rightarrow> 'a list" where |
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"postorder \<langle>\<rangle> = []" | |
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"postorder \<langle>l, x, r\<rangle> = postorder l @ postorder r @ [x]" |
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||
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text\<open>Binary Search Tree:\<close> |
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fun bst_wrt :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a tree \<Rightarrow> bool" where |
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"bst_wrt P \<langle>\<rangle> \<longleftrightarrow> True" | |
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"bst_wrt P \<langle>l, a, r\<rangle> \<longleftrightarrow> |
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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)" |
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abbreviation bst :: "('a::linorder) tree \<Rightarrow> bool" where |
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"bst \<equiv> bst_wrt (<)" |
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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 map_tree}\<close> |
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lemma eq_map_tree_Leaf[simp]: "map_tree f t = Leaf \<longleftrightarrow> t = Leaf" |
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by (rule tree.map_disc_iff) |
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lemma eq_Leaf_map_tree[simp]: "Leaf = map_tree f t \<longleftrightarrow> t = Leaf" |
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by (cases t) auto |
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subsection \<open>@{const size}\<close> |
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||
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lemma size1_size: "size1 t = size t + 1" |
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by (induction t) simp_all |
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lemma size1_ge0[simp]: "0 < size1 t" |
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by (simp add: size1_size) |
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lemma eq_size_0[simp]: "size t = 0 \<longleftrightarrow> t = Leaf" |
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by(cases t) auto |
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lemma eq_0_size[simp]: "0 = size t \<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 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_size) |
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subsection \<open>@{const set_tree}\<close> |
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lemma eq_set_tree_empty[simp]: "set_tree t = {} \<longleftrightarrow> t = Leaf" |
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by (cases t) auto |
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lemma eq_empty_set_tree[simp]: "{} = set_tree t \<longleftrightarrow> t = Leaf" |
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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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subsection \<open>@{const subtrees}\<close> |
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lemma neq_subtrees_empty[simp]: "subtrees t \<noteq> {}" |
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by (cases t)(auto) |
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lemma neq_empty_subtrees[simp]: "{} \<noteq> subtrees t" |
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by (cases t)(auto) |
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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 eq_height_0[simp]: "height t = 0 \<longleftrightarrow> t = Leaf" |
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by(cases t) auto |
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lemma eq_0_height[simp]: "0 = height t \<longleftrightarrow> t = Leaf" |
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by(cases t) auto |
165 |
||
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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_size] by(arith) |
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lemma height_subtrees: "s \<in> subtrees t \<Longrightarrow> height s \<le> height t" |
197 |
by (induction t) auto |
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lemma min_height_le_height: "min_height t \<le> height t" |
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by(induction t) auto |
202 |
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lemma min_height_map_tree[simp]: "min_height (map_tree f t) = min_height t" |
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by (induction t) auto |
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lemma min_height_size1: "2 ^ min_height t \<le> size1 t" |
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proof(induction t) |
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case (Node l a r) |
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have "(2::nat) ^ min_height (Node l a r) \<le> 2 ^ min_height l + 2 ^ min_height r" |
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by (simp add: min_def) |
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also have "\<dots> \<le> size1(Node l a r)" using Node.IH by simp |
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finally show ?case . |
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qed simp |
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subsection \<open>@{const complete}\<close> |
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lemma complete_iff_height: "complete t \<longleftrightarrow> (min_height t = height t)" |
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apply(induction t) |
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apply simp |
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apply (simp add: min_def max_def) |
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by (metis le_antisym le_trans min_height_le_height) |
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lemma size1_if_complete: "complete t \<Longrightarrow> size1 t = 2 ^ height t" |
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by (induction t) auto |
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lemma size_if_complete: "complete t \<Longrightarrow> size t = 2 ^ height t - 1" |
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using size1_if_complete[simplified size1_size] by fastforce |
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lemma size1_height_if_incomplete: |
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"\<not> complete t \<Longrightarrow> size1 t < 2 ^ height t" |
|
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proof(induction t) |
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case Leaf thus ?case by simp |
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63770 | 234 |
next |
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case (Node l x r) |
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have 1: ?case if h: "height l < height r" |
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using h size1_height[of l] size1_height[of r] power_strict_increasing[OF h, of "2::nat"] |
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by(auto simp: max_def simp del: power_strict_increasing_iff) |
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have 2: ?case if h: "height l > height r" |
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using h size1_height[of l] size1_height[of r] power_strict_increasing[OF h, of "2::nat"] |
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by(auto simp: max_def simp del: power_strict_increasing_iff) |
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have 3: ?case if h: "height l = height r" and c: "\<not> complete l" |
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using h size1_height[of r] Node.IH(1)[OF c] by(simp) |
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have 4: ?case if h: "height l = height r" and c: "\<not> complete r" |
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using h size1_height[of l] Node.IH(2)[OF c] by(simp) |
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from 1 2 3 4 Node.prems show ?case apply (simp add: max_def) by linarith |
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qed |
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lemma complete_iff_min_height: "complete t \<longleftrightarrow> (height t = min_height t)" |
250 |
by(auto simp add: complete_iff_height) |
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||
252 |
lemma min_height_size1_if_incomplete: |
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"\<not> complete t \<Longrightarrow> 2 ^ min_height t < size1 t" |
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proof(induction t) |
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255 |
case Leaf thus ?case by simp |
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256 |
next |
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case (Node l x r) |
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have 1: ?case if h: "min_height l < min_height r" |
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using h min_height_size1[of l] min_height_size1[of r] power_strict_increasing[OF h, of "2::nat"] |
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by(auto simp: max_def simp del: power_strict_increasing_iff) |
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have 2: ?case if h: "min_height l > min_height r" |
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using h min_height_size1[of l] min_height_size1[of r] power_strict_increasing[OF h, of "2::nat"] |
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263 |
by(auto simp: max_def simp del: power_strict_increasing_iff) |
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have 3: ?case if h: "min_height l = min_height r" and c: "\<not> complete l" |
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using h min_height_size1[of r] Node.IH(1)[OF c] by(simp add: complete_iff_min_height) |
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have 4: ?case if h: "min_height l = min_height r" and c: "\<not> complete r" |
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using h min_height_size1[of l] Node.IH(2)[OF c] by(simp add: complete_iff_min_height) |
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from 1 2 3 4 Node.prems show ?case |
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by (fastforce simp: complete_iff_min_height[THEN iffD1]) |
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qed |
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||
272 |
lemma complete_if_size1_height: "size1 t = 2 ^ height t \<Longrightarrow> complete t" |
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273 |
using size1_height_if_incomplete by fastforce |
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|
64533 | 275 |
lemma complete_if_size1_min_height: "size1 t = 2 ^ min_height t \<Longrightarrow> complete t" |
69117 | 276 |
using min_height_size1_if_incomplete by fastforce |
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64533 | 278 |
lemma complete_iff_size1: "complete t \<longleftrightarrow> size1 t = 2 ^ height t" |
279 |
using complete_if_size1_height size1_if_complete by blast |
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280 |
||
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63861 | 282 |
subsection \<open>@{const balanced}\<close> |
283 |
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284 |
lemma balanced_subtreeL: "balanced (Node l x r) \<Longrightarrow> balanced l" |
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285 |
by(simp add: balanced_def) |
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286 |
|
63861 | 287 |
lemma balanced_subtreeR: "balanced (Node l x r) \<Longrightarrow> balanced r" |
288 |
by(simp add: balanced_def) |
|
289 |
||
290 |
lemma balanced_subtrees: "\<lbrakk> balanced t; s \<in> subtrees t \<rbrakk> \<Longrightarrow> balanced s" |
|
291 |
using [[simp_depth_limit=1]] |
|
292 |
by(induction t arbitrary: s) |
|
293 |
(auto simp add: balanced_subtreeL balanced_subtreeR) |
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text\<open>Balanced trees have optimal height:\<close> |
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lemma balanced_optimal: |
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fixes t :: "'a tree" and t' :: "'b tree" |
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assumes "balanced t" "size t \<le> size t'" shows "height t \<le> height t'" |
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proof (cases "complete t") |
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case True |
64924 | 302 |
have "(2::nat) ^ height t \<le> 2 ^ height t'" |
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303 |
proof - |
64924 | 304 |
have "2 ^ height t = size1 t" |
69115 | 305 |
using True by (simp add: size1_if_complete) |
68998 | 306 |
also have "\<dots> \<le> size1 t'" using assms(2) by(simp add: size1_size) |
64924 | 307 |
also have "\<dots> \<le> 2 ^ height t'" by (rule size1_height) |
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308 |
finally show ?thesis . |
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qed |
64924 | 310 |
thus ?thesis by (simp) |
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next |
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case False |
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have "(2::nat) ^ min_height t < 2 ^ height t'" |
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314 |
proof - |
64533 | 315 |
have "(2::nat) ^ min_height t < size1 t" |
316 |
by(rule min_height_size1_if_incomplete[OF False]) |
|
68998 | 317 |
also have "\<dots> \<le> size1 t'" using assms(2) by (simp add: size1_size) |
64918 | 318 |
also have "\<dots> \<le> 2 ^ height t'" by(rule size1_height) |
319 |
finally have "(2::nat) ^ min_height t < (2::nat) ^ height t'" . |
|
64924 | 320 |
thus ?thesis . |
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321 |
qed |
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322 |
hence *: "min_height t < height t'" by simp |
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323 |
have "min_height t + 1 = height t" |
64540 | 324 |
using min_height_le_height[of t] assms(1) False |
63829 | 325 |
by (simp add: complete_iff_height balanced_def) |
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326 |
with * show ?thesis by arith |
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327 |
qed |
63036 | 328 |
|
329 |
||
63861 | 330 |
subsection \<open>@{const wbalanced}\<close> |
331 |
||
332 |
lemma wbalanced_subtrees: "\<lbrakk> wbalanced t; s \<in> subtrees t \<rbrakk> \<Longrightarrow> wbalanced s" |
|
333 |
using [[simp_depth_limit=1]] by(induction t arbitrary: s) auto |
|
334 |
||
335 |
||
64887 | 336 |
subsection \<open>@{const ipl}\<close> |
63413 | 337 |
|
338 |
text \<open>The internal path length of a tree:\<close> |
|
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||
64923 | 340 |
lemma ipl_if_complete_int: |
341 |
"complete t \<Longrightarrow> int(ipl t) = (int(height t) - 2) * 2^(height t) + 2" |
|
342 |
apply(induction t) |
|
343 |
apply simp |
|
344 |
apply simp |
|
345 |
apply (simp add: algebra_simps size_if_complete of_nat_diff) |
|
346 |
done |
|
63413 | 347 |
|
348 |
||
59776 | 349 |
subsection "List of entries" |
350 |
||
65340 | 351 |
lemma eq_inorder_Nil[simp]: "inorder t = [] \<longleftrightarrow> t = Leaf" |
65339 | 352 |
by (cases t) auto |
353 |
||
65340 | 354 |
lemma eq_Nil_inorder[simp]: "[] = inorder t \<longleftrightarrow> t = Leaf" |
65339 | 355 |
by (cases t) auto |
356 |
||
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|
357 |
lemma set_inorder[simp]: "set (inorder t) = set_tree t" |
58424 | 358 |
by (induction t) auto |
57250 | 359 |
|
59776 | 360 |
lemma set_preorder[simp]: "set (preorder t) = set_tree t" |
361 |
by (induction t) auto |
|
362 |
||
64925 | 363 |
lemma set_postorder[simp]: "set (postorder t) = set_tree t" |
364 |
by (induction t) auto |
|
365 |
||
59776 | 366 |
lemma length_preorder[simp]: "length (preorder t) = size t" |
367 |
by (induction t) auto |
|
368 |
||
369 |
lemma length_inorder[simp]: "length (inorder t) = size t" |
|
370 |
by (induction t) auto |
|
371 |
||
64925 | 372 |
lemma length_postorder[simp]: "length (postorder t) = size t" |
373 |
by (induction t) auto |
|
374 |
||
59776 | 375 |
lemma preorder_map: "preorder (map_tree f t) = map f (preorder t)" |
376 |
by (induction t) auto |
|
377 |
||
378 |
lemma inorder_map: "inorder (map_tree f t) = map f (inorder t)" |
|
379 |
by (induction t) auto |
|
380 |
||
64925 | 381 |
lemma postorder_map: "postorder (map_tree f t) = map f (postorder t)" |
382 |
by (induction t) auto |
|
383 |
||
63765 | 384 |
lemma inorder2_inorder: "inorder2 t xs = inorder t @ xs" |
385 |
by (induction t arbitrary: xs) auto |
|
386 |
||
57687 | 387 |
|
63861 | 388 |
subsection \<open>Binary Search Tree\<close> |
59561 | 389 |
|
66606 | 390 |
lemma bst_wrt_mono: "(\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> bst_wrt P t \<Longrightarrow> bst_wrt Q t" |
59928 | 391 |
by (induction t) (auto) |
392 |
||
67399 | 393 |
lemma bst_wrt_le_if_bst: "bst t \<Longrightarrow> bst_wrt (\<le>) t" |
66606 | 394 |
using bst_wrt_mono less_imp_le by blast |
395 |
||
67399 | 396 |
lemma bst_wrt_le_iff_sorted: "bst_wrt (\<le>) t \<longleftrightarrow> sorted (inorder t)" |
59561 | 397 |
apply (induction t) |
398 |
apply(simp) |
|
68109 | 399 |
by (fastforce simp: sorted_append intro: less_imp_le less_trans) |
59561 | 400 |
|
67399 | 401 |
lemma bst_iff_sorted_wrt_less: "bst t \<longleftrightarrow> sorted_wrt (<) (inorder t)" |
59928 | 402 |
apply (induction t) |
403 |
apply simp |
|
68109 | 404 |
apply (fastforce simp: sorted_wrt_append) |
59928 | 405 |
done |
406 |
||
59776 | 407 |
|
63861 | 408 |
subsection \<open>@{const heap}\<close> |
60505 | 409 |
|
410 |
||
63861 | 411 |
subsection \<open>@{const mirror}\<close> |
59561 | 412 |
|
413 |
lemma mirror_Leaf[simp]: "mirror t = \<langle>\<rangle> \<longleftrightarrow> t = \<langle>\<rangle>" |
|
414 |
by (induction t) simp_all |
|
415 |
||
65339 | 416 |
lemma Leaf_mirror[simp]: "\<langle>\<rangle> = mirror t \<longleftrightarrow> t = \<langle>\<rangle>" |
417 |
using mirror_Leaf by fastforce |
|
418 |
||
59561 | 419 |
lemma size_mirror[simp]: "size(mirror t) = size t" |
420 |
by (induction t) simp_all |
|
421 |
||
422 |
lemma size1_mirror[simp]: "size1(mirror t) = size1 t" |
|
68998 | 423 |
by (simp add: size1_size) |
59561 | 424 |
|
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diff
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425 |
lemma height_mirror[simp]: "height(mirror t) = height t" |
59776 | 426 |
by (induction t) simp_all |
427 |
||
66659 | 428 |
lemma min_height_mirror [simp]: "min_height (mirror t) = min_height t" |
429 |
by (induction t) simp_all |
|
430 |
||
431 |
lemma ipl_mirror [simp]: "ipl (mirror t) = ipl t" |
|
432 |
by (induction t) simp_all |
|
433 |
||
59776 | 434 |
lemma inorder_mirror: "inorder(mirror t) = rev(inorder t)" |
435 |
by (induction t) simp_all |
|
436 |
||
437 |
lemma map_mirror: "map_tree f (mirror t) = mirror (map_tree f t)" |
|
438 |
by (induction t) simp_all |
|
439 |
||
59561 | 440 |
lemma mirror_mirror[simp]: "mirror(mirror t) = t" |
441 |
by (induction t) simp_all |
|
442 |
||
57250 | 443 |
end |