author  nipkow 
Thu, 01 Sep 2016 15:57:54 +0200  
changeset 63755  182c111190e5 
parent 63665  15f48ce7ec23 
child 63765  e60020520b15 
permissions  rwrr 
57250  1 
(* Author: Tobias Nipkow *) 
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(* Todo: 
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size t = 2^h  1 \<Longrightarrow> complete t 
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(min_)height of balanced trees via floorlog 
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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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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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where 
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"left Leaf = Leaf" 
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 "right Leaf = Leaf" 
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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> 
58438  22 

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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 complete_size1: "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 complete_size1[simplified size1_def] by fastforce 
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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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qed 
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qed 
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qed 
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subsection \<open>Balanced\<close> 
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abbreviation "balanced t \<equiv> (height t  min_height t \<le> 1)" 
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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 
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have "(2::nat) ^ height t  1 \<le> 2 ^ height t'  1" 
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proof  
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have "(2::nat) ^ height t  1 = size t" 
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using True by (simp add: complete_iff_height size_if_complete) 
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also note assms(2) 
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also have "size t' \<le> 2 ^ height t'  1" by (rule size_height) 
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finally show ?thesis . 
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qed 
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thus ?thesis by (simp add: le_diff_iff) 
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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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proof  
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have "(2::nat) ^ min_height t \<le> size t" 
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by(rule min_height_le_size_if_incomplete[OF False]) 
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also note assms(2) 
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also have "size t' \<le> 2 ^ height t'  1" by(rule size_height) 
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finally show ?thesis 
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using power_eq_0_iff[of "2::nat" "height t'"] by linarith 
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qed 
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hence *: "min_height t < height t'" by simp 
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have "min_height t + 1 = height t" 
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using min_hight_le_height[of t] assms(1) False 
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by (simp add: complete_iff_height) 
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with * show ?thesis by arith 
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qed 
63036  224 

225 

63413  226 
subsection \<open>Path length\<close> 
227 

228 
text \<open>The internal path length of a tree:\<close> 

229 

230 
fun path_len :: "'a tree \<Rightarrow> nat" where 

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"path_len Leaf = 0 "  

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"path_len (Node l _ r) = path_len l + size l + path_len r + size r" 

233 

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lemma path_len_if_bal: "complete t 
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\<Longrightarrow> path_len t = (let n = height t in 2 + n*2^n  2^(n+1))" 
236 
proof(induction t) 

237 
case (Node l x r) 

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have *: "2^(n+1) \<le> 2 + n*2^n" for n :: nat 

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by(induction n) auto 

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have **: "(0::nat) < 2^n" for n :: nat by simp 

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let ?h = "height r" 

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show ?case using Node *[of ?h] **[of ?h] by (simp add: size_if_complete Let_def) 
63413  243 
qed simp 
244 

245 

57687  246 
subsection "The set of subtrees" 
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57250  248 
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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58424  252 
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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57450  255 
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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58424  258 
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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59776  262 
subsection "List of entries" 
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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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lemma set_inorder[simp]: "set (inorder t) = set_tree t" 
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by (induction t) auto 
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lemma set_preorder[simp]: "set (preorder t) = set_tree t" 
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by (induction t) auto 

277 

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lemma length_preorder[simp]: "length (preorder t) = size t" 

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by (induction t) auto 

280 

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lemma length_inorder[simp]: "length (inorder t) = size t" 

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by (induction t) auto 

283 

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lemma preorder_map: "preorder (map_tree f t) = map f (preorder t)" 

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by (induction t) auto 

286 

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lemma inorder_map: "inorder (map_tree f t) = map f (inorder t)" 

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by (induction t) auto 

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60500  291 
subsection \<open>Binary Search Tree predicate\<close> 
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57450  293 
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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60500  297 
text\<open>In case there are duplicates:\<close> 
59561  298 

299 
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)" 

303 

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lemma (in linorder) bst_eq_if_bst: "bst t \<Longrightarrow> bst_eq t" 
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by (induction t) (auto) 

306 

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lemma (in linorder) bst_eq_imp_sorted: "bst_eq t \<Longrightarrow> sorted (inorder t)" 
308 
apply (induction t) 

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apply(simp) 

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by (fastforce simp: sorted_append sorted_Cons intro: less_imp_le less_trans) 

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lemma (in linorder) distinct_preorder_if_bst: "bst t \<Longrightarrow> distinct (preorder t)" 
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apply (induction t) 

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apply simp 

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apply(fastforce elim: order.asym) 

316 
done 

317 

318 
lemma (in linorder) distinct_inorder_if_bst: "bst t \<Longrightarrow> distinct (inorder t)" 

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apply (induction t) 

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apply simp 

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apply(fastforce elim: order.asym) 

322 
done 

323 

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60505  325 
subsection "The heap predicate" 
326 

327 
fun heap :: "'a::linorder tree \<Rightarrow> bool" where 

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"heap Leaf = True"  

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"heap (Node l m r) = 

330 
(heap l \<and> heap r \<and> (\<forall>x \<in> set_tree l \<union> set_tree r. m \<le> x))" 

331 

332 

61585  333 
subsection "Function \<open>mirror\<close>" 
59561  334 

335 
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>" 

338 

339 
lemma mirror_Leaf[simp]: "mirror t = \<langle>\<rangle> \<longleftrightarrow> t = \<langle>\<rangle>" 

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by (induction t) simp_all 

341 

342 
lemma size_mirror[simp]: "size(mirror t) = size t" 

343 
by (induction t) simp_all 

344 

345 
lemma size1_mirror[simp]: "size1(mirror t) = size1 t" 

346 
by (simp add: size1_def) 

347 

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lemma height_mirror[simp]: "height(mirror t) = height t" 
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by (induction t) simp_all 
350 

351 
lemma inorder_mirror: "inorder(mirror t) = rev(inorder t)" 

352 
by (induction t) simp_all 

353 

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lemma map_mirror: "map_tree f (mirror t) = mirror (map_tree f t)" 

355 
by (induction t) simp_all 

356 

59561  357 
lemma mirror_mirror[simp]: "mirror(mirror t) = t" 
358 
by (induction t) simp_all 

359 

57250  360 
end 