new theory of binary trees

--- a/src/HOL/Library/Library.thy	Thu Jun 12 18:02:39 2014 +0200
+++ b/src/HOL/Library/Library.thy	Thu Jun 12 21:23:28 2014 +0200
@@ -64,6 +64,7 @@
   Sublist
   Sum_of_Squares
   Transitive_Closure_Table
+  Tree
   While_Combinator
 begin
 end

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Tree.thy	Thu Jun 12 21:23:28 2014 +0200
@@ -0,0 +1,46 @@
+(* Author: Tobias Nipkow *)
+
+header {* Binary Tree *}
+
+theory Tree
+imports Main
+begin
+
+datatype 'a tree = Leaf | Node "'a tree" 'a "'a tree"
+
+fun set_tree :: "'a tree \<Rightarrow> 'a set" where
+"set_tree Leaf = {}" |
+"set_tree (Node l x r) = insert x (set_tree l \<union> set_tree r)"
+
+fun subtrees :: "'a tree \<Rightarrow> 'a tree set" where
+"subtrees Leaf = {Leaf}" |
+"subtrees (Node l a r) = insert (Node l a r) (subtrees l \<union> subtrees r)"
+
+fun inorder :: "'a tree \<Rightarrow> 'a list" where
+"inorder Leaf = []" |
+"inorder (Node l x r) = inorder l @ [x] @ inorder r"
+
+text{* Binary Search Tree predicate: *}
+fun bst :: "'a::linorder tree \<Rightarrow> bool" where
+"bst Leaf = True" |
+"bst (Node l a r) =
+  (bst l & bst r & (\<forall>x \<in> set_tree l. x < a)  & (\<forall>x \<in> set_tree r. a < x))"
+
+lemma neq_Leaf_iff: "(t \<noteq> Leaf) = (\<exists>l a r. t = Node l a r)"
+by (cases t) auto
+
+lemma set_inorder[simp]: "set(inorder t) = set_tree t"
+by (induction t) auto
+
+lemma set_treeE: "a : set_tree t \<Longrightarrow> \<exists>l r. Node l a r \<in> subtrees t"
+by(induction t)(auto)
+
+lemma Node_notin_subtrees_if[simp]:
+  "a \<notin> set_tree t \<Longrightarrow> Node l a r \<notin> subtrees t"
+by (induction t) auto
+
+lemma in_set_tree_if:
+  "Node l a r \<in> subtrees t \<Longrightarrow> a \<in> set_tree t"
+by (metis Node_notin_subtrees_if)
+
+end