author | nipkow |
Fri, 09 Sep 2016 14:15:16 +0200 | |
changeset 63829 | 6a05c8cbf7de |
parent 63770 | a67397b13eb5 |
child 63861 | 90360390a916 |
permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow *) |
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(* Todo: |
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(min_)height of balanced trees via floorlog |
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minimal path_len of balanced trees |
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*) |
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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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is_Leaf: Leaf ("\<langle>\<rangle>") | |
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Node (left: "'a tree") (val: 'a) (right: "'a tree") ("(1\<langle>_,/ _,/ _\<rangle>)") |
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Library/Tree: use datatype_new, bst is an inductive predicate
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where |
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"left Leaf = Leaf" |
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| "right Leaf = Leaf" |
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register tree with datatype_compat ot support QuickCheck
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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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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 "The Height" |
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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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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 size1_height: "size t + 1 \<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 "size(Node l a r) + 1 = (size l + 1) + (size r + 1)" by simp |
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also have "size l + 1 \<le> 2 ^ height l" by(rule Node.IH(1)) |
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also have "size r + 1 \<le> 2 ^ height r" by(rule Node.IH(2)) |
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also have "(2::nat) ^ height l \<le> 2 ^ height r" using True by simp |
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finally show ?thesis using True by (auto simp: max_def mult_2) |
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next |
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case False |
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have "size(Node l a r) + 1 = (size l + 1) + (size r + 1)" by simp |
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also have "size l + 1 \<le> 2 ^ height l" by(rule Node.IH(1)) |
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also have "size r + 1 \<le> 2 ^ height r" by(rule Node.IH(2)) |
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also have "(2::nat) ^ height r \<le> 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] by(arith) |
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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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lemma min_hight_le_height: "min_height t \<le> height t" |
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by(induction t) auto |
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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_le_size1: "2 ^ min_height t \<le> size t + 1" |
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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> size(Node l a r) + 1" using Node.IH by simp |
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finally show ?case . |
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qed simp |
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subsection \<open>Complete\<close> |
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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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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_hight_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_def] by fastforce |
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lemma complete_if_size: "size t = 2 ^ height t - 1 \<Longrightarrow> complete t" |
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proof (induct "height t" arbitrary: t) |
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case 0 thus ?case by (simp add: size_0_iff_Leaf) |
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next |
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case (Suc h) |
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hence "t \<noteq> Leaf" by auto |
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then obtain l a r where [simp]: "t = Node l a r" |
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by (auto simp: neq_Leaf_iff) |
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have 1: "height l \<le> h" and 2: "height r \<le> h" using Suc(2) by(auto) |
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have 3: "~ height l < h" |
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proof |
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assume 0: "height l < h" |
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have "size t = size l + (size r + 1)" by simp |
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also note size_height[of l] |
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also note size1_height[of r] |
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also have "(2::nat) ^ height l - 1 < 2 ^ h - 1" |
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using 0 by (simp add: diff_less_mono) |
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also have "(2::nat) ^ height r \<le> 2 ^ h" using 2 by simp |
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also have "(2::nat) ^ h - 1 + 2 ^ h = 2 ^ (Suc h) - 1" by (simp) |
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also have "\<dots> = size t" using Suc(2,3) by simp |
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finally show False by (simp add: diff_le_mono) |
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qed |
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have 4: "~ height r < h" |
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proof |
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assume 0: "height r < h" |
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have "size t = (size l + 1) + size r" by simp |
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also note size_height[of r] |
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also note size1_height[of l] |
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also have "(2::nat) ^ height r - 1 < 2 ^ h - 1" |
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using 0 by (simp add: diff_less_mono) |
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also have "(2::nat) ^ height l \<le> 2 ^ h" using 1 by simp |
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also have "(2::nat) ^ h + (2 ^ h - 1) = 2 ^ (Suc h) - 1" by (simp) |
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also have "\<dots> = size t" using Suc(2,3) by simp |
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finally show False by (simp add: diff_le_mono) |
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qed |
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from 1 2 3 4 have *: "height l = h" "height r = h" by linarith+ |
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hence "size l = 2 ^ height l - 1" "size r = 2 ^ height r - 1" |
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using Suc(3) size_height[of l] size_height[of r] by (auto) |
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with * Suc(1) show ?case by simp |
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qed |
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lemma complete_iff_size: "complete t \<longleftrightarrow> size t = 2 ^ height t - 1" |
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using complete_if_size size_if_complete by blast |
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text\<open>A better lower bound for incomplete trees:\<close> |
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lemma min_height_le_size_if_incomplete: |
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"\<not> complete t \<Longrightarrow> 2 ^ min_height t \<le> size t" |
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proof(induction t) |
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case Leaf thus ?case by simp |
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next |
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case (Node l a r) |
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show ?case (is "?l \<le> ?r") |
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proof (cases "complete l") |
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case l: True thus ?thesis |
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proof (cases "complete r") |
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case r: True |
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have "height l \<noteq> height r" using Node.prems l r by simp |
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hence "?l < 2 ^ min_height l + 2 ^ min_height r" |
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using l r by (simp add: min_def complete_iff_height) |
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also have "\<dots> = (size l + 1) + (size r + 1)" |
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using l r size_if_complete[where ?'a = 'a] |
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by (simp add: complete_iff_height) |
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also have "\<dots> \<le> ?r + 1" by simp |
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finally show ?thesis by arith |
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next |
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case r: False |
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have "?l \<le> 2 ^ min_height l + 2 ^ min_height r" by (simp add: min_def) |
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also have "\<dots> \<le> size l + 1 + size r" |
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using Node.IH(2)[OF r] l size_if_complete[where ?'a = 'a] |
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by (simp add: complete_iff_height) |
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also have "\<dots> = ?r" by simp |
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finally show ?thesis . |
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qed |
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next |
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case l: False thus ?thesis |
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proof (cases "complete r") |
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case r: True |
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have "?l \<le> 2 ^ min_height l + 2 ^ min_height r" by (simp add: min_def) |
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also have "\<dots> \<le> size l + (size r + 1)" |
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using Node.IH(1)[OF l] r size_if_complete[where ?'a = 'a] |
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by (simp add: complete_iff_height) |
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also have "\<dots> = ?r" by simp |
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finally show ?thesis . |
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next |
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case r: False |
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have "?l \<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> size l + size r" |
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using Node.IH(1)[OF l] Node.IH(2)[OF r] by (simp) |
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also have "\<dots> \<le> ?r" by simp |
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finally show ?thesis . |
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226 |
qed |
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227 |
qed |
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|
228 |
qed |
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229 |
|
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230 |
|
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231 |
subsection \<open>Balanced\<close> |
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232 |
|
63829 | 233 |
definition balanced :: "'a tree \<Rightarrow> bool" where |
234 |
"balanced t = (height t - min_height t \<le> 1)" |
|
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235 |
|
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236 |
text\<open>Balanced trees have optimal height:\<close> |
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237 |
|
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|
238 |
lemma balanced_optimal: |
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|
239 |
fixes t :: "'a tree" and t' :: "'b tree" |
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240 |
assumes "balanced t" "size t \<le> size t'" shows "height t \<le> height t'" |
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|
241 |
proof (cases "complete t") |
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|
242 |
case True |
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243 |
have "(2::nat) ^ height t - 1 \<le> 2 ^ height t' - 1" |
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|
244 |
proof - |
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|
245 |
have "(2::nat) ^ height t - 1 = size t" |
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246 |
using True by (simp add: complete_iff_height size_if_complete) |
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|
247 |
also note assms(2) |
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|
248 |
also have "size t' \<le> 2 ^ height t' - 1" by (rule size_height) |
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|
249 |
finally show ?thesis . |
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|
250 |
qed |
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|
251 |
thus ?thesis by (simp add: le_diff_iff) |
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|
252 |
next |
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|
253 |
case False |
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|
254 |
have "(2::nat) ^ min_height t < 2 ^ height t'" |
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|
255 |
proof - |
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|
256 |
have "(2::nat) ^ min_height t \<le> size t" |
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|
257 |
by(rule min_height_le_size_if_incomplete[OF False]) |
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|
258 |
also note assms(2) |
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|
259 |
also have "size t' \<le> 2 ^ height t' - 1" by(rule size_height) |
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|
260 |
finally show ?thesis |
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|
261 |
using power_eq_0_iff[of "2::nat" "height t'"] by linarith |
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|
262 |
qed |
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|
263 |
hence *: "min_height t < height t'" by simp |
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|
264 |
have "min_height t + 1 = height t" |
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|
265 |
using min_hight_le_height[of t] assms(1) False |
63829 | 266 |
by (simp add: complete_iff_height balanced_def) |
63755
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|
267 |
with * show ?thesis by arith |
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|
268 |
qed |
63036 | 269 |
|
270 |
||
63413 | 271 |
subsection \<open>Path length\<close> |
272 |
||
273 |
text \<open>The internal path length of a tree:\<close> |
|
274 |
||
275 |
fun path_len :: "'a tree \<Rightarrow> nat" where |
|
276 |
"path_len Leaf = 0 " | |
|
277 |
"path_len (Node l _ r) = path_len l + size l + path_len r + size r" |
|
278 |
||
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279 |
lemma path_len_if_bal: "complete t |
63413 | 280 |
\<Longrightarrow> path_len t = (let n = height t in 2 + n*2^n - 2^(n+1))" |
281 |
proof(induction t) |
|
282 |
case (Node l x r) |
|
283 |
have *: "2^(n+1) \<le> 2 + n*2^n" for n :: nat |
|
284 |
by(induction n) auto |
|
285 |
have **: "(0::nat) < 2^n" for n :: nat by simp |
|
286 |
let ?h = "height r" |
|
63755
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287 |
show ?case using Node *[of ?h] **[of ?h] by (simp add: size_if_complete Let_def) |
63413 | 288 |
qed simp |
289 |
||
290 |
||
57687 | 291 |
subsection "The set of subtrees" |
292 |
||
57250 | 293 |
fun subtrees :: "'a tree \<Rightarrow> 'a tree set" where |
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294 |
"subtrees \<langle>\<rangle> = {\<langle>\<rangle>}" | |
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|
295 |
"subtrees (\<langle>l, a, r\<rangle>) = insert \<langle>l, a, r\<rangle> (subtrees l \<union> subtrees r)" |
57250 | 296 |
|
58424 | 297 |
lemma set_treeE: "a \<in> set_tree t \<Longrightarrow> \<exists>l r. \<langle>l, a, r\<rangle> \<in> subtrees t" |
298 |
by (induction t)(auto) |
|
57449
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hoelzl
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|
299 |
|
57450 | 300 |
lemma Node_notin_subtrees_if[simp]: "a \<notin> set_tree t \<Longrightarrow> Node l a r \<notin> subtrees t" |
58424 | 301 |
by (induction t) auto |
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hoelzl
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|
302 |
|
58424 | 303 |
lemma in_set_tree_if: "\<langle>l, a, r\<rangle> \<in> subtrees t \<Longrightarrow> a \<in> set_tree t" |
304 |
by (metis Node_notin_subtrees_if) |
|
57449
f81da03b9ebd
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hoelzl
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changeset
|
305 |
|
57687 | 306 |
|
59776 | 307 |
subsection "List of entries" |
308 |
||
309 |
fun preorder :: "'a tree \<Rightarrow> 'a list" where |
|
310 |
"preorder \<langle>\<rangle> = []" | |
|
311 |
"preorder \<langle>l, x, r\<rangle> = x # preorder l @ preorder r" |
|
57687 | 312 |
|
57250 | 313 |
fun inorder :: "'a tree \<Rightarrow> 'a list" where |
58424 | 314 |
"inorder \<langle>\<rangle> = []" | |
315 |
"inorder \<langle>l, x, r\<rangle> = inorder l @ [x] @ inorder r" |
|
57250 | 316 |
|
63765 | 317 |
text\<open>A linear version avoiding append:\<close> |
318 |
fun inorder2 :: "'a tree \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
|
319 |
"inorder2 \<langle>\<rangle> xs = xs" | |
|
320 |
"inorder2 \<langle>l, x, r\<rangle> xs = inorder2 l (x # inorder2 r xs)" |
|
321 |
||
322 |
||
57449
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hoelzl
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changeset
|
323 |
lemma set_inorder[simp]: "set (inorder t) = set_tree t" |
58424 | 324 |
by (induction t) auto |
57250 | 325 |
|
59776 | 326 |
lemma set_preorder[simp]: "set (preorder t) = set_tree t" |
327 |
by (induction t) auto |
|
328 |
||
329 |
lemma length_preorder[simp]: "length (preorder t) = size t" |
|
330 |
by (induction t) auto |
|
331 |
||
332 |
lemma length_inorder[simp]: "length (inorder t) = size t" |
|
333 |
by (induction t) auto |
|
334 |
||
335 |
lemma preorder_map: "preorder (map_tree f t) = map f (preorder t)" |
|
336 |
by (induction t) auto |
|
337 |
||
338 |
lemma inorder_map: "inorder (map_tree f t) = map f (inorder t)" |
|
339 |
by (induction t) auto |
|
340 |
||
63765 | 341 |
lemma inorder2_inorder: "inorder2 t xs = inorder t @ xs" |
342 |
by (induction t arbitrary: xs) auto |
|
343 |
||
57687 | 344 |
|
60500 | 345 |
subsection \<open>Binary Search Tree predicate\<close> |
57250 | 346 |
|
57450 | 347 |
fun (in linorder) bst :: "'a tree \<Rightarrow> bool" where |
58424 | 348 |
"bst \<langle>\<rangle> \<longleftrightarrow> True" | |
349 |
"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)" |
|
57250 | 350 |
|
60500 | 351 |
text\<open>In case there are duplicates:\<close> |
59561 | 352 |
|
353 |
fun (in linorder) bst_eq :: "'a tree \<Rightarrow> bool" where |
|
354 |
"bst_eq \<langle>\<rangle> \<longleftrightarrow> True" | |
|
355 |
"bst_eq \<langle>l,a,r\<rangle> \<longleftrightarrow> |
|
356 |
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)" |
|
357 |
||
59928 | 358 |
lemma (in linorder) bst_eq_if_bst: "bst t \<Longrightarrow> bst_eq t" |
359 |
by (induction t) (auto) |
|
360 |
||
59561 | 361 |
lemma (in linorder) bst_eq_imp_sorted: "bst_eq t \<Longrightarrow> sorted (inorder t)" |
362 |
apply (induction t) |
|
363 |
apply(simp) |
|
364 |
by (fastforce simp: sorted_append sorted_Cons intro: less_imp_le less_trans) |
|
365 |
||
59928 | 366 |
lemma (in linorder) distinct_preorder_if_bst: "bst t \<Longrightarrow> distinct (preorder t)" |
367 |
apply (induction t) |
|
368 |
apply simp |
|
369 |
apply(fastforce elim: order.asym) |
|
370 |
done |
|
371 |
||
372 |
lemma (in linorder) distinct_inorder_if_bst: "bst t \<Longrightarrow> distinct (inorder t)" |
|
373 |
apply (induction t) |
|
374 |
apply simp |
|
375 |
apply(fastforce elim: order.asym) |
|
376 |
done |
|
377 |
||
59776 | 378 |
|
60505 | 379 |
subsection "The heap predicate" |
380 |
||
381 |
fun heap :: "'a::linorder tree \<Rightarrow> bool" where |
|
382 |
"heap Leaf = True" | |
|
383 |
"heap (Node l m r) = |
|
384 |
(heap l \<and> heap r \<and> (\<forall>x \<in> set_tree l \<union> set_tree r. m \<le> x))" |
|
385 |
||
386 |
||
61585 | 387 |
subsection "Function \<open>mirror\<close>" |
59561 | 388 |
|
389 |
fun mirror :: "'a tree \<Rightarrow> 'a tree" where |
|
390 |
"mirror \<langle>\<rangle> = Leaf" | |
|
391 |
"mirror \<langle>l,x,r\<rangle> = \<langle>mirror r, x, mirror l\<rangle>" |
|
392 |
||
393 |
lemma mirror_Leaf[simp]: "mirror t = \<langle>\<rangle> \<longleftrightarrow> t = \<langle>\<rangle>" |
|
394 |
by (induction t) simp_all |
|
395 |
||
396 |
lemma size_mirror[simp]: "size(mirror t) = size t" |
|
397 |
by (induction t) simp_all |
|
398 |
||
399 |
lemma size1_mirror[simp]: "size1(mirror t) = size1 t" |
|
400 |
by (simp add: size1_def) |
|
401 |
||
60808
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nipkow
parents:
60507
diff
changeset
|
402 |
lemma height_mirror[simp]: "height(mirror t) = height t" |
59776 | 403 |
by (induction t) simp_all |
404 |
||
405 |
lemma inorder_mirror: "inorder(mirror t) = rev(inorder t)" |
|
406 |
by (induction t) simp_all |
|
407 |
||
408 |
lemma map_mirror: "map_tree f (mirror t) = mirror (map_tree f t)" |
|
409 |
by (induction t) simp_all |
|
410 |
||
59561 | 411 |
lemma mirror_mirror[simp]: "mirror(mirror t) = t" |
412 |
by (induction t) simp_all |
|
413 |
||
57250 | 414 |
end |