(* Author: Tobias Nipkow *)
header {* Binary Tree *}
theory Tree
imports Main
begin
datatype_new 'a tree = Leaf | Node (left: "'a tree") (val: 'a) (right: "'a tree")
where
"left Leaf = Leaf"
| "right Leaf = Leaf"
lemma neq_Leaf_iff: "(t \<noteq> Leaf) = (\<exists>l a r. t = Node l a r)"
by (cases t) auto
lemma set_tree_Node2: "set_tree(Node l x r) = insert x (set_tree l \<union> set_tree r)"
by auto
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)"
lemma set_treeE: "a \<in> 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)
fun inorder :: "'a tree \<Rightarrow> 'a list" where
"inorder Leaf = []" |
"inorder (Node l x r) = inorder l @ [x] @ inorder r"
lemma set_inorder[simp]: "set (inorder t) = set_tree t"
by (induction t) auto
subsection {* Binary Search Tree predicate *}
fun (in linorder) bst :: "'a tree \<Rightarrow> bool" where
"bst Leaf \<longleftrightarrow> True" |
"bst (Node l a r) \<longleftrightarrow> bst l \<and> bst r \<and> (\<forall>x\<in>set_tree l. x < a) \<and> (\<forall>x\<in>set_tree r. a < x)"
lemma (in linorder) bst_imp_sorted: "bst t \<Longrightarrow> sorted (inorder t)"
by (induction t) (auto simp: sorted_append sorted_Cons intro: less_imp_le less_trans)
end