src/HOL/Library/Tree.thy
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(* Author: Tobias Nipkow *)
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section {* Binary Tree *}
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theory Tree
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imports Main
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begin
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datatype 'a tree =
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  Leaf ("\<langle>\<rangle>") |
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  Node (left: "'a tree") (val: 'a) (right: "'a tree") ("\<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{* Can be seen as counting the number of leaves rather than nodes: *}
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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 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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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 "Inorder list of entries"
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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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subsection {* Binary Search Tree predicate *}
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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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lemma (in linorder) bst_imp_sorted: "bst t \<Longrightarrow> sorted (inorder t)"
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by (induction t) (auto simp: sorted_append sorted_Cons intro: less_imp_le less_trans)
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text{* In case there are duplicates: *}
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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_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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subsection "Function @{text mirror}"
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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 mirror_mirror[simp]: "mirror(mirror t) = t"
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by (induction t) simp_all
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subsection "Deletion of the rightmost entry"
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fun del_rightmost :: "'a tree \<Rightarrow> 'a tree * 'a" where
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"del_rightmost \<langle>l, a, \<langle>\<rangle>\<rangle> = (l,a)" |
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"del_rightmost \<langle>l, a, r\<rangle> = (let (r',x) = del_rightmost r in (\<langle>l, a, r'\<rangle>, x))"
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lemma del_rightmost_set_tree_if_bst:
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  "\<lbrakk> del_rightmost t = (t',x); bst t; t \<noteq> Leaf \<rbrakk>
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  \<Longrightarrow> x \<in> set_tree t \<and> set_tree t' = set_tree t - {x}"
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apply(induction t arbitrary: t' rule: del_rightmost.induct)
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  apply (fastforce simp: ball_Un split: prod.splits)+
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done
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lemma del_rightmost_set_tree:
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  "\<lbrakk> del_rightmost t = (t',x);  t \<noteq> \<langle>\<rangle> \<rbrakk> \<Longrightarrow> set_tree t = insert x (set_tree t')"
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apply(induction t arbitrary: t' rule: del_rightmost.induct)
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by (auto split: prod.splits) auto
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lemma del_rightmost_bst:
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  "\<lbrakk> del_rightmost t = (t',x);  bst t;  t \<noteq> \<langle>\<rangle> \<rbrakk> \<Longrightarrow> bst t'"
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proof(induction t arbitrary: t' rule: del_rightmost.induct)
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  case (2 l a rl b rr)
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  let ?r = "Node rl b rr"
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  from "2.prems"(1) obtain r' where 1: "del_rightmost ?r = (r',x)" and [simp]: "t' = Node l a r'"
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    by(simp split: prod.splits)
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  from "2.prems"(2) 1 del_rightmost_set_tree[OF 1] show ?case by(auto)(simp add: "2.IH")
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qed auto
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lemma del_rightmost_greater: "\<lbrakk> del_rightmost t = (t',x);  bst t;  t \<noteq> \<langle>\<rangle> \<rbrakk>
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  \<Longrightarrow> \<forall>a\<in>set_tree t'. a < x"
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proof(induction t arbitrary: t' rule: del_rightmost.induct)
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  case (2 l a rl b rr)
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  from "2.prems"(1) obtain r'
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  where dm: "del_rightmost (Node rl b rr) = (r',x)" and [simp]: "t' = Node l a r'"
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    by(simp split: prod.splits)
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  show ?case using "2.prems"(2) "2.IH"[OF dm] del_rightmost_set_tree_if_bst[OF dm]
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    by (fastforce simp add: ball_Un)
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qed simp_all
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lemma del_rightmost_Max:
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  "\<lbrakk> del_rightmost t = (t',x);  bst t;  t \<noteq> \<langle>\<rangle> \<rbrakk> \<Longrightarrow> x = Max(set_tree t)"
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by (metis Max_insert2 del_rightmost_greater del_rightmost_set_tree finite_set_tree less_le_not_le)
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end