(* Author: Tobias Nipkow *)
section \<open>Binary Tree\<close>
theory Tree
imports Main
begin
datatype 'a tree =
Leaf ("\<langle>\<rangle>") |
Node (left: "'a tree") (val: 'a) (right: "'a tree") ("\<langle>_, _, _\<rangle>")
where
"left Leaf = Leaf"
| "right Leaf = Leaf"
datatype_compat tree
text\<open>Can be seen as counting the number of leaves rather than nodes:\<close>
definition size1 :: "'a tree \<Rightarrow> nat" where
"size1 t = size t + 1"
lemma size1_simps[simp]:
"size1 \<langle>\<rangle> = 1"
"size1 \<langle>l, x, r\<rangle> = size1 l + size1 r"
by (simp_all add: size1_def)
lemma size_0_iff_Leaf: "size t = 0 \<longleftrightarrow> t = Leaf"
by(cases t) auto
lemma neq_Leaf_iff: "(t \<noteq> \<langle>\<rangle>) = (\<exists>l a r. t = \<langle>l, a, r\<rangle>)"
by (cases t) auto
lemma finite_set_tree[simp]: "finite(set_tree t)"
by(induction t) auto
lemma size_map_tree[simp]: "size (map_tree f t) = size t"
by (induction t) auto
lemma size1_map_tree[simp]: "size1 (map_tree f t) = size1 t"
by (simp add: size1_def)
subsection "The depth"
fun depth :: "'a tree => nat" where
"depth Leaf = 0" |
"depth (Node t1 a t2) = Suc (max (depth t1) (depth t2))"
lemma depth_map_tree[simp]: "depth (map_tree f t) = depth t"
by (induction t) auto
subsection "The set of subtrees"
fun subtrees :: "'a tree \<Rightarrow> 'a tree set" where
"subtrees \<langle>\<rangle> = {\<langle>\<rangle>}" |
"subtrees (\<langle>l, a, r\<rangle>) = insert \<langle>l, a, r\<rangle> (subtrees l \<union> subtrees r)"
lemma set_treeE: "a \<in> set_tree t \<Longrightarrow> \<exists>l r. \<langle>l, a, r\<rangle> \<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: "\<langle>l, a, r\<rangle> \<in> subtrees t \<Longrightarrow> a \<in> set_tree t"
by (metis Node_notin_subtrees_if)
subsection "List of entries"
fun preorder :: "'a tree \<Rightarrow> 'a list" where
"preorder \<langle>\<rangle> = []" |
"preorder \<langle>l, x, r\<rangle> = x # preorder l @ preorder r"
fun inorder :: "'a tree \<Rightarrow> 'a list" where
"inorder \<langle>\<rangle> = []" |
"inorder \<langle>l, x, r\<rangle> = inorder l @ [x] @ inorder r"
lemma set_inorder[simp]: "set (inorder t) = set_tree t"
by (induction t) auto
lemma set_preorder[simp]: "set (preorder t) = set_tree t"
by (induction t) auto
lemma length_preorder[simp]: "length (preorder t) = size t"
by (induction t) auto
lemma length_inorder[simp]: "length (inorder t) = size t"
by (induction t) auto
lemma preorder_map: "preorder (map_tree f t) = map f (preorder t)"
by (induction t) auto
lemma inorder_map: "inorder (map_tree f t) = map f (inorder t)"
by (induction t) auto
subsection \<open>Binary Search Tree predicate\<close>
fun (in linorder) bst :: "'a tree \<Rightarrow> bool" where
"bst \<langle>\<rangle> \<longleftrightarrow> True" |
"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)"
text\<open>In case there are duplicates:\<close>
fun (in linorder) bst_eq :: "'a tree \<Rightarrow> bool" where
"bst_eq \<langle>\<rangle> \<longleftrightarrow> True" |
"bst_eq \<langle>l,a,r\<rangle> \<longleftrightarrow>
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)"
lemma (in linorder) bst_eq_if_bst: "bst t \<Longrightarrow> bst_eq t"
by (induction t) (auto)
lemma (in linorder) bst_eq_imp_sorted: "bst_eq t \<Longrightarrow> sorted (inorder t)"
apply (induction t)
apply(simp)
by (fastforce simp: sorted_append sorted_Cons intro: less_imp_le less_trans)
lemma (in linorder) distinct_preorder_if_bst: "bst t \<Longrightarrow> distinct (preorder t)"
apply (induction t)
apply simp
apply(fastforce elim: order.asym)
done
lemma (in linorder) distinct_inorder_if_bst: "bst t \<Longrightarrow> distinct (inorder t)"
apply (induction t)
apply simp
apply(fastforce elim: order.asym)
done
subsection "The heap predicate"
fun heap :: "'a::linorder tree \<Rightarrow> bool" where
"heap Leaf = True" |
"heap (Node l m r) =
(heap l \<and> heap r \<and> (\<forall>x \<in> set_tree l \<union> set_tree r. m \<le> x))"
subsection "Function @{text mirror}"
fun mirror :: "'a tree \<Rightarrow> 'a tree" where
"mirror \<langle>\<rangle> = Leaf" |
"mirror \<langle>l,x,r\<rangle> = \<langle>mirror r, x, mirror l\<rangle>"
lemma mirror_Leaf[simp]: "mirror t = \<langle>\<rangle> \<longleftrightarrow> t = \<langle>\<rangle>"
by (induction t) simp_all
lemma size_mirror[simp]: "size(mirror t) = size t"
by (induction t) simp_all
lemma size1_mirror[simp]: "size1(mirror t) = size1 t"
by (simp add: size1_def)
lemma depth_mirror[simp]: "depth(mirror t) = depth t"
by (induction t) simp_all
lemma inorder_mirror: "inorder(mirror t) = rev(inorder t)"
by (induction t) simp_all
lemma map_mirror: "map_tree f (mirror t) = mirror (map_tree f t)"
by (induction t) simp_all
lemma mirror_mirror[simp]: "mirror(mirror t) = t"
by (induction t) simp_all
subsection "Deletion of the rightmost entry"
fun del_rightmost :: "'a tree \<Rightarrow> 'a tree * 'a" where
"del_rightmost \<langle>l, a, \<langle>\<rangle>\<rangle> = (l,a)" |
"del_rightmost \<langle>l, a, r\<rangle> = (let (r',x) = del_rightmost r in (\<langle>l, a, r'\<rangle>, x))"
lemma del_rightmost_set_tree_if_bst:
"\<lbrakk> del_rightmost t = (t',x); bst t; t \<noteq> Leaf \<rbrakk>
\<Longrightarrow> x \<in> set_tree t \<and> set_tree t' = set_tree t - {x}"
apply(induction t arbitrary: t' rule: del_rightmost.induct)
apply (fastforce simp: ball_Un split: prod.splits)+
done
lemma del_rightmost_set_tree:
"\<lbrakk> del_rightmost t = (t',x); t \<noteq> \<langle>\<rangle> \<rbrakk> \<Longrightarrow> set_tree t = insert x (set_tree t')"
apply(induction t arbitrary: t' rule: del_rightmost.induct)
by (auto split: prod.splits) auto
lemma del_rightmost_bst:
"\<lbrakk> del_rightmost t = (t',x); bst t; t \<noteq> \<langle>\<rangle> \<rbrakk> \<Longrightarrow> bst t'"
proof(induction t arbitrary: t' rule: del_rightmost.induct)
case (2 l a rl b rr)
let ?r = "Node rl b rr"
from "2.prems"(1) obtain r' where 1: "del_rightmost ?r = (r',x)" and [simp]: "t' = Node l a r'"
by(simp split: prod.splits)
from "2.prems"(2) 1 del_rightmost_set_tree[OF 1] show ?case by(auto)(simp add: "2.IH")
qed auto
lemma del_rightmost_greater: "\<lbrakk> del_rightmost t = (t',x); bst t; t \<noteq> \<langle>\<rangle> \<rbrakk>
\<Longrightarrow> \<forall>a\<in>set_tree t'. a < x"
proof(induction t arbitrary: t' rule: del_rightmost.induct)
case (2 l a rl b rr)
from "2.prems"(1) obtain r'
where dm: "del_rightmost (Node rl b rr) = (r',x)" and [simp]: "t' = Node l a r'"
by(simp split: prod.splits)
show ?case using "2.prems"(2) "2.IH"[OF dm] del_rightmost_set_tree_if_bst[OF dm]
by (fastforce simp add: ball_Un)
qed simp_all
lemma del_rightmost_Max:
"\<lbrakk> del_rightmost t = (t',x); bst t; t \<noteq> \<langle>\<rangle> \<rbrakk> \<Longrightarrow> x = Max(set_tree t)"
by (metis Max_insert2 del_rightmost_greater del_rightmost_set_tree finite_set_tree less_le_not_le)
end