(* Author: Tobias Nipkow *)
(* Todo: minimal ipl of balanced trees *)
section \<open>Binary Tree\<close>
theory Tree
imports Main
begin
datatype 'a tree =
Leaf ("\<langle>\<rangle>") |
Node "'a tree" (root_val: 'a) "'a tree" ("(1\<langle>_,/ _,/ _\<rangle>)")
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"
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)"
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>"
class height = fixes height :: "'a \<Rightarrow> nat"
instantiation tree :: (type)height
begin
fun height_tree :: "'a tree => nat" where
"height Leaf = 0" |
"height (Node t1 a t2) = max (height t1) (height t2) + 1"
instance ..
end
fun min_height :: "'a tree \<Rightarrow> nat" where
"min_height Leaf = 0" |
"min_height (Node l _ r) = min (min_height l) (min_height r) + 1"
fun complete :: "'a tree \<Rightarrow> bool" where
"complete Leaf = True" |
"complete (Node l x r) = (complete l \<and> complete r \<and> height l = height r)"
definition balanced :: "'a tree \<Rightarrow> bool" where
"balanced t = (height t - min_height t \<le> 1)"
text \<open>Weight balanced:\<close>
fun wbalanced :: "'a tree \<Rightarrow> bool" where
"wbalanced Leaf = True" |
"wbalanced (Node l x r) = (abs(int(size l) - int(size r)) \<le> 1 \<and> wbalanced l \<and> wbalanced r)"
text \<open>Internal path length:\<close>
fun ipl :: "'a tree \<Rightarrow> nat" where
"ipl Leaf = 0 " |
"ipl (Node l _ r) = ipl l + size l + ipl r + size r"
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"
text\<open>A linear version avoiding append:\<close>
fun inorder2 :: "'a tree \<Rightarrow> 'a list \<Rightarrow> 'a list" where
"inorder2 \<langle>\<rangle> xs = xs" |
"inorder2 \<langle>l, x, r\<rangle> xs = inorder2 l (x # inorder2 r xs)"
fun postorder :: "'a tree \<Rightarrow> 'a list" where
"postorder \<langle>\<rangle> = []" |
"postorder \<langle>l, x, r\<rangle> = postorder l @ postorder r @ [x]"
text\<open>Binary Search Tree:\<close>
fun bst_wrt :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a tree \<Rightarrow> bool" where
"bst_wrt P \<langle>\<rangle> \<longleftrightarrow> True" |
"bst_wrt P \<langle>l, a, r\<rangle> \<longleftrightarrow>
bst_wrt P l \<and> bst_wrt P r \<and> (\<forall>x\<in>set_tree l. P x a) \<and> (\<forall>x\<in>set_tree r. P a x)"
abbreviation bst :: "('a::linorder) tree \<Rightarrow> bool" where
"bst \<equiv> bst_wrt (op <)"
fun (in linorder) heap :: "'a 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 \<open>@{const map_tree}\<close>
lemma eq_map_tree_Leaf[simp]: "map_tree f t = Leaf \<longleftrightarrow> t = Leaf"
by (rule tree.map_disc_iff)
lemma eq_Leaf_map_tree[simp]: "Leaf = map_tree f t \<longleftrightarrow> t = Leaf"
by (cases t) auto
subsection \<open>@{const size}\<close>
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 size1_ge0[simp]: "0 < size1 t"
by (simp add: size1_def)
lemma eq_size_0[simp]: "size t = 0 \<longleftrightarrow> t = Leaf"
by(cases t) auto
lemma eq_0_size[simp]: "0 = size t \<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 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 \<open>@{const set_tree}\<close>
lemma eq_set_tree_empty[simp]: "set_tree t = {} \<longleftrightarrow> t = Leaf"
by (cases t) auto
lemma eq_empty_set_tree[simp]: "{} = set_tree t \<longleftrightarrow> t = Leaf"
by (cases t) auto
lemma finite_set_tree[simp]: "finite(set_tree t)"
by(induction t) auto
subsection \<open>@{const subtrees}\<close>
lemma neq_subtrees_empty[simp]: "subtrees t \<noteq> {}"
by (cases t)(auto)
lemma neq_empty_subtrees[simp]: "{} \<noteq> subtrees t"
by (cases t)(auto)
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 \<open>@{const height} and @{const min_height}\<close>
lemma eq_height_0[simp]: "height t = 0 \<longleftrightarrow> t = Leaf"
by(cases t) auto
lemma eq_0_height[simp]: "0 = height t \<longleftrightarrow> t = Leaf"
by(cases t) auto
lemma height_map_tree[simp]: "height (map_tree f t) = height t"
by (induction t) auto
lemma height_le_size_tree: "height t \<le> size (t::'a tree)"
by (induction t) auto
lemma size1_height: "size1 t \<le> 2 ^ height (t::'a tree)"
proof(induction t)
case (Node l a r)
show ?case
proof (cases "height l \<le> height r")
case True
have "size1(Node l a r) = size1 l + size1 r" by simp
also have "\<dots> \<le> 2 ^ height l + 2 ^ height r" using Node.IH by arith
also have "\<dots> \<le> 2 ^ height r + 2 ^ height r" using True by simp
also have "\<dots> = 2 ^ height (Node l a r)"
using True by (auto simp: max_def mult_2)
finally show ?thesis .
next
case False
have "size1(Node l a r) = size1 l + size1 r" by simp
also have "\<dots> \<le> 2 ^ height l + 2 ^ height r" using Node.IH by arith
also have "\<dots> \<le> 2 ^ height l + 2 ^ height l" using False by simp
finally show ?thesis using False by (auto simp: max_def mult_2)
qed
qed simp
corollary size_height: "size t \<le> 2 ^ height (t::'a tree) - 1"
using size1_height[of t, unfolded size1_def] by(arith)
lemma height_subtrees: "s \<in> subtrees t \<Longrightarrow> height s \<le> height t"
by (induction t) auto
lemma min_height_le_height: "min_height t \<le> height t"
by(induction t) auto
lemma min_height_map_tree[simp]: "min_height (map_tree f t) = min_height t"
by (induction t) auto
lemma min_height_size1: "2 ^ min_height t \<le> size1 t"
proof(induction t)
case (Node l a r)
have "(2::nat) ^ min_height (Node l a r) \<le> 2 ^ min_height l + 2 ^ min_height r"
by (simp add: min_def)
also have "\<dots> \<le> size1(Node l a r)" using Node.IH by simp
finally show ?case .
qed simp
subsection \<open>@{const complete}\<close>
lemma complete_iff_height: "complete t \<longleftrightarrow> (min_height t = height t)"
apply(induction t)
apply simp
apply (simp add: min_def max_def)
by (metis le_antisym le_trans min_height_le_height)
lemma size1_if_complete: "complete t \<Longrightarrow> size1 t = 2 ^ height t"
by (induction t) auto
lemma size_if_complete: "complete t \<Longrightarrow> size t = 2 ^ height t - 1"
using size1_if_complete[simplified size1_def] by fastforce
lemma complete_if_size1_height: "size1 t = 2 ^ height t \<Longrightarrow> complete t"
proof (induct "height t" arbitrary: t)
case 0 thus ?case by (simp)
next
case (Suc h)
hence "t \<noteq> Leaf" by auto
then obtain l a r where [simp]: "t = Node l a r"
by (auto simp: neq_Leaf_iff)
have 1: "height l \<le> h" and 2: "height r \<le> h" using Suc(2) by(auto)
have 3: "\<not> height l < h"
proof
assume 0: "height l < h"
have "size1 t = size1 l + size1 r" by simp
also have "\<dots> \<le> 2 ^ height l + 2 ^ height r"
using size1_height[of l] size1_height[of r] by arith
also have " \<dots> < 2 ^ h + 2 ^ height r" using 0 by (simp)
also have " \<dots> \<le> 2 ^ h + 2 ^ h" using 2 by (simp)
also have "\<dots> = 2 ^ (Suc h)" by (simp)
also have "\<dots> = size1 t" using Suc(2,3) by simp
finally have "size1 t < size1 t" .
thus False by (simp)
qed
have 4: "\<not> height r < h"
proof
assume 0: "height r < h"
have "size1 t = size1 l + size1 r" by simp
also have "\<dots> \<le> 2 ^ height l + 2 ^ height r"
using size1_height[of l] size1_height[of r] by arith
also have " \<dots> < 2 ^ height l + 2 ^ h" using 0 by (simp)
also have " \<dots> \<le> 2 ^ h + 2 ^ h" using 1 by (simp)
also have "\<dots> = 2 ^ (Suc h)" by (simp)
also have "\<dots> = size1 t" using Suc(2,3) by simp
finally have "size1 t < size1 t" .
thus False by (simp)
qed
from 1 2 3 4 have *: "height l = h" "height r = h" by linarith+
hence "size1 l = 2 ^ height l" "size1 r = 2 ^ height r"
using Suc(3) size1_height[of l] size1_height[of r] by (auto)
with * Suc(1) show ?case by simp
qed
text\<open>The following proof involves \<open>\<ge>\<close>/\<open>>\<close> chains rather than the standard
\<open>\<le>\<close>/\<open><\<close> chains. To chain the elements together the transitivity rules \<open>xtrans\<close>
are used.\<close>
lemma complete_if_size1_min_height: "size1 t = 2 ^ min_height t \<Longrightarrow> complete t"
proof (induct "min_height t" arbitrary: t)
case 0 thus ?case by (simp add: size1_def)
next
case (Suc h)
hence "t \<noteq> Leaf" by auto
then obtain l a r where [simp]: "t = Node l a r"
by (auto simp: neq_Leaf_iff)
have 1: "h \<le> min_height l" and 2: "h \<le> min_height r" using Suc(2) by(auto)
have 3: "\<not> h < min_height l"
proof
assume 0: "h < min_height l"
have "size1 t = size1 l + size1 r" by simp
also note min_height_size1[of l]
also(xtrans) note min_height_size1[of r]
also(xtrans) have "(2::nat) ^ min_height l > 2 ^ h"
using 0 by (simp add: diff_less_mono)
also(xtrans) have "(2::nat) ^ min_height r \<ge> 2 ^ h" using 2 by simp
also(xtrans) have "(2::nat) ^ h + 2 ^ h = 2 ^ (Suc h)" by (simp)
also have "\<dots> = size1 t" using Suc(2,3) by simp
finally show False by (simp add: diff_le_mono)
qed
have 4: "\<not> h < min_height r"
proof
assume 0: "h < min_height r"
have "size1 t = size1 l + size1 r" by simp
also note min_height_size1[of l]
also(xtrans) note min_height_size1[of r]
also(xtrans) have "(2::nat) ^ min_height r > 2 ^ h"
using 0 by (simp add: diff_less_mono)
also(xtrans) have "(2::nat) ^ min_height l \<ge> 2 ^ h" using 1 by simp
also(xtrans) have "(2::nat) ^ h + 2 ^ h = 2 ^ (Suc h)" by (simp)
also have "\<dots> = size1 t" using Suc(2,3) by simp
finally show False by (simp add: diff_le_mono)
qed
from 1 2 3 4 have *: "min_height l = h" "min_height r = h" by linarith+
hence "size1 l = 2 ^ min_height l" "size1 r = 2 ^ min_height r"
using Suc(3) min_height_size1[of l] min_height_size1[of r] by (auto)
with * Suc(1) show ?case
by (simp add: complete_iff_height)
qed
lemma complete_iff_size1: "complete t \<longleftrightarrow> size1 t = 2 ^ height t"
using complete_if_size1_height size1_if_complete by blast
text\<open>Better bounds for incomplete trees:\<close>
lemma size1_height_if_incomplete:
"\<not> complete t \<Longrightarrow> size1 t < 2 ^ height t"
by (meson antisym_conv complete_iff_size1 not_le size1_height)
lemma min_height_size1_if_incomplete:
"\<not> complete t \<Longrightarrow> 2 ^ min_height t < size1 t"
by (metis complete_if_size1_min_height le_less min_height_size1)
subsection \<open>@{const balanced}\<close>
lemma balanced_subtreeL: "balanced (Node l x r) \<Longrightarrow> balanced l"
by(simp add: balanced_def)
lemma balanced_subtreeR: "balanced (Node l x r) \<Longrightarrow> balanced r"
by(simp add: balanced_def)
lemma balanced_subtrees: "\<lbrakk> balanced t; s \<in> subtrees t \<rbrakk> \<Longrightarrow> balanced s"
using [[simp_depth_limit=1]]
by(induction t arbitrary: s)
(auto simp add: balanced_subtreeL balanced_subtreeR)
text\<open>Balanced trees have optimal height:\<close>
lemma balanced_optimal:
fixes t :: "'a tree" and t' :: "'b tree"
assumes "balanced t" "size t \<le> size t'" shows "height t \<le> height t'"
proof (cases "complete t")
case True
have "(2::nat) ^ height t \<le> 2 ^ height t'"
proof -
have "2 ^ height t = size1 t"
using True by (simp add: complete_iff_height size1_if_complete)
also have "\<dots> \<le> size1 t'" using assms(2) by(simp add: size1_def)
also have "\<dots> \<le> 2 ^ height t'" by (rule size1_height)
finally show ?thesis .
qed
thus ?thesis by (simp)
next
case False
have "(2::nat) ^ min_height t < 2 ^ height t'"
proof -
have "(2::nat) ^ min_height t < size1 t"
by(rule min_height_size1_if_incomplete[OF False])
also have "\<dots> \<le> size1 t'" using assms(2) by (simp add: size1_def)
also have "\<dots> \<le> 2 ^ height t'" by(rule size1_height)
finally have "(2::nat) ^ min_height t < (2::nat) ^ height t'" .
thus ?thesis .
qed
hence *: "min_height t < height t'" by simp
have "min_height t + 1 = height t"
using min_height_le_height[of t] assms(1) False
by (simp add: complete_iff_height balanced_def)
with * show ?thesis by arith
qed
subsection \<open>@{const wbalanced}\<close>
lemma wbalanced_subtrees: "\<lbrakk> wbalanced t; s \<in> subtrees t \<rbrakk> \<Longrightarrow> wbalanced s"
using [[simp_depth_limit=1]] by(induction t arbitrary: s) auto
subsection \<open>@{const ipl}\<close>
text \<open>The internal path length of a tree:\<close>
lemma ipl_if_complete_int:
"complete t \<Longrightarrow> int(ipl t) = (int(height t) - 2) * 2^(height t) + 2"
apply(induction t)
apply simp
apply simp
apply (simp add: algebra_simps size_if_complete of_nat_diff)
done
subsection "List of entries"
lemma eq_inorder_Nil[simp]: "inorder t = [] \<longleftrightarrow> t = Leaf"
by (cases t) auto
lemma eq_Nil_inorder[simp]: "[] = inorder t \<longleftrightarrow> t = Leaf"
by (cases t) auto
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 set_postorder[simp]: "set (postorder 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 length_postorder[simp]: "length (postorder 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
lemma postorder_map: "postorder (map_tree f t) = map f (postorder t)"
by (induction t) auto
lemma inorder2_inorder: "inorder2 t xs = inorder t @ xs"
by (induction t arbitrary: xs) auto
subsection \<open>Binary Search Tree\<close>
lemma bst_wrt_mono: "(\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> bst_wrt P t \<Longrightarrow> bst_wrt Q t"
by (induction t) (auto)
lemma bst_wrt_le_if_bst: "bst t \<Longrightarrow> bst_wrt (op \<le>) t"
using bst_wrt_mono less_imp_le by blast
lemma bst_wrt_le_iff_sorted: "bst_wrt (op \<le>) t \<longleftrightarrow> sorted (inorder t)"
apply (induction t)
apply(simp)
by (fastforce simp: sorted_append sorted_Cons intro: less_imp_le less_trans)
lemma bst_iff_sorted_wrt_less: "bst t \<longleftrightarrow> sorted_wrt (op <) (inorder t)"
apply (induction t)
apply simp
apply (fastforce simp: sorted_wrt_Cons sorted_wrt_append)
done
subsection \<open>@{const heap}\<close>
subsection \<open>@{const mirror}\<close>
lemma mirror_Leaf[simp]: "mirror t = \<langle>\<rangle> \<longleftrightarrow> t = \<langle>\<rangle>"
by (induction t) simp_all
lemma Leaf_mirror[simp]: "\<langle>\<rangle> = mirror t \<longleftrightarrow> t = \<langle>\<rangle>"
using mirror_Leaf by fastforce
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 height_mirror[simp]: "height(mirror t) = height t"
by (induction t) simp_all
lemma min_height_mirror [simp]: "min_height (mirror t) = min_height t"
by (induction t) simp_all
lemma ipl_mirror [simp]: "ipl (mirror t) = ipl 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
end