src/HOL/Library/Tree.thy
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added tree lemmas
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(* Author: Tobias Nipkow *)
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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>
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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_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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subsection "The set of subtrees"
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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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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 "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
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lemma length_preorder[simp]: "length (preorder t) = size t"
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by (induction t) auto
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lemma length_inorder[simp]: "length (inorder t) = size t"
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by (induction t) auto
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lemma preorder_map: "preorder (map_tree f t) = map f (preorder t)"
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by (induction t) auto
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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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subsection \<open>Binary Search Tree predicate\<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>In case there are 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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lemma (in linorder) bst_eq_if_bst: "bst t \<Longrightarrow> bst_eq t"
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by (induction t) (auto)
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lemma (in linorder) bst_eq_imp_sorted: "bst_eq t \<Longrightarrow> sorted (inorder t)"
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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)
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done
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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)
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done
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subsection "The heap predicate"
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fun heap :: "'a::linorder 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 "Function \<open>mirror\<close>"
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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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lemma mirror_Leaf[simp]: "mirror t = \<langle>\<rangle> \<longleftrightarrow> t = \<langle>\<rangle>"
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by (induction t) simp_all
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lemma size_mirror[simp]: "size(mirror t) = size t"
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by (induction t) simp_all
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lemma size1_mirror[simp]: "size1(mirror t) = size1 t"
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by (simp add: size1_def)
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lemma height_mirror[simp]: "height(mirror t) = height t"
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by (induction t) simp_all
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lemma inorder_mirror: "inorder(mirror t) = rev(inorder t)"
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by (induction t) simp_all
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lemma map_mirror: "map_tree f (mirror t) = mirror (map_tree f t)"
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by (induction t) simp_all
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lemma mirror_mirror[simp]: "mirror(mirror t) = t"
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by (induction t) simp_all
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end