# Theory Tree

```(* Author: Tobias Nipkow *)
(* Todo: minimal ipl of almost complete trees *)

section ‹Binary Tree›

theory Tree
imports Main
begin

datatype 'a tree =
Leaf ("⟨⟩") |
Node "'a tree" ("value": 'a) "'a tree" ("(1⟨_,/ _,/ _⟩)")
datatype_compat tree

primrec left :: "'a tree ⇒ 'a tree" where
"left (Node l v r) = l" |
"left Leaf = Leaf"

primrec right :: "'a tree ⇒ 'a tree" where
"right (Node l v r) = r" |
"right Leaf = Leaf"

text‹Counting the number of leaves rather than nodes:›

fun size1 :: "'a tree ⇒ nat" where
"size1 ⟨⟩ = 1" |
"size1 ⟨l, x, r⟩ = size1 l + size1 r"

fun subtrees :: "'a tree ⇒ 'a tree set" where
"subtrees ⟨⟩ = {⟨⟩}" |
"subtrees (⟨l, a, r⟩) = {⟨l, a, r⟩} ∪ subtrees l ∪ subtrees r"

fun mirror :: "'a tree ⇒ 'a tree" where
"mirror ⟨⟩ = Leaf" |
"mirror ⟨l,x,r⟩ = ⟨mirror r, x, mirror l⟩"

class height = fixes height :: "'a ⇒ nat"

instantiation tree :: (type)height
begin

fun height_tree :: "'a tree => nat" where
"height Leaf = 0" |
"height (Node l a r) = max (height l) (height r) + 1"

instance ..

end

fun min_height :: "'a tree ⇒ nat" where
"min_height Leaf = 0" |
"min_height (Node l _ r) = min (min_height l) (min_height r) + 1"

fun complete :: "'a tree ⇒ bool" where
"complete Leaf = True" |
"complete (Node l x r) = (height l = height r ∧ complete l ∧ complete r)"

text ‹Almost complete:›
definition acomplete :: "'a tree ⇒ bool" where
"acomplete t = (height t - min_height t ≤ 1)"

text ‹Weight balanced:›
fun wbalanced :: "'a tree ⇒ bool" where
"wbalanced Leaf = True" |
"wbalanced (Node l x r) = (abs(int(size l) - int(size r)) ≤ 1 ∧ wbalanced l ∧ wbalanced r)"

text ‹Internal path length:›
fun ipl :: "'a tree ⇒ nat" where
"ipl Leaf = 0 " |
"ipl (Node l _ r) = ipl l + size l + ipl r + size r"

fun preorder :: "'a tree ⇒ 'a list" where
"preorder ⟨⟩ = []" |
"preorder ⟨l, x, r⟩ = x # preorder l @ preorder r"

fun inorder :: "'a tree ⇒ 'a list" where
"inorder ⟨⟩ = []" |
"inorder ⟨l, x, r⟩ = inorder l @ [x] @ inorder r"

text‹A linear version avoiding append:›
fun inorder2 :: "'a tree ⇒ 'a list ⇒ 'a list" where
"inorder2 ⟨⟩ xs = xs" |
"inorder2 ⟨l, x, r⟩ xs = inorder2 l (x # inorder2 r xs)"

fun postorder :: "'a tree ⇒ 'a list" where
"postorder ⟨⟩ = []" |
"postorder ⟨l, x, r⟩ = postorder l @ postorder r @ [x]"

text‹Binary Search Tree:›
fun bst_wrt :: "('a ⇒ 'a ⇒ bool) ⇒ 'a tree ⇒ bool" where
"bst_wrt P ⟨⟩ ⟷ True" |
"bst_wrt P ⟨l, a, r⟩ ⟷
(∀x∈set_tree l. P x a) ∧ (∀x∈set_tree r. P a x) ∧ bst_wrt P l ∧ bst_wrt P r"

abbreviation bst :: "('a::linorder) tree ⇒ bool" where
"bst ≡ bst_wrt (<)"

fun (in linorder) heap :: "'a tree ⇒ bool" where
"heap Leaf = True" |
"heap (Node l m r) =
((∀x ∈ set_tree l ∪ set_tree r. m ≤ x) ∧ heap l ∧ heap r)"

subsection ‹\<^const>‹map_tree››

lemma eq_map_tree_Leaf[simp]: "map_tree f t = Leaf ⟷ t = Leaf"
by (rule tree.map_disc_iff)

lemma eq_Leaf_map_tree[simp]: "Leaf = map_tree f t ⟷ t = Leaf"
by (cases t) auto

subsection ‹\<^const>‹size››

lemma size1_size: "size1 t = size t + 1"
by (induction t) simp_all

lemma size1_ge0[simp]: "0 < size1 t"
by (simp add: size1_size)

lemma eq_size_0[simp]: "size t = 0 ⟷ t = Leaf"
by(cases t) auto

lemma eq_0_size[simp]: "0 = size t ⟷ t = Leaf"
by(cases t) auto

lemma neq_Leaf_iff: "(t ≠ ⟨⟩) = (∃l a r. t = ⟨l, a, r⟩)"
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_size)

subsection ‹\<^const>‹set_tree››

lemma eq_set_tree_empty[simp]: "set_tree t = {} ⟷ t = Leaf"
by (cases t) auto

lemma eq_empty_set_tree[simp]: "{} = set_tree t ⟷ t = Leaf"
by (cases t) auto

lemma finite_set_tree[simp]: "finite(set_tree t)"
by(induction t) auto

subsection ‹\<^const>‹subtrees››

lemma neq_subtrees_empty[simp]: "subtrees t ≠ {}"
by (cases t)(auto)

lemma neq_empty_subtrees[simp]: "{} ≠ subtrees t"
by (cases t)(auto)

lemma size_subtrees: "s ∈ subtrees t ⟹ size s ≤ size t"
by(induction t)(auto)

lemma set_treeE: "a ∈ set_tree t ⟹ ∃l r. ⟨l, a, r⟩ ∈ subtrees t"
by (induction t)(auto)

lemma Node_notin_subtrees_if[simp]: "a ∉ set_tree t ⟹ Node l a r ∉ subtrees t"
by (induction t) auto

lemma in_set_tree_if: "⟨l, a, r⟩ ∈ subtrees t ⟹ a ∈ set_tree t"
by (metis Node_notin_subtrees_if)

subsection ‹\<^const>‹height› and \<^const>‹min_height››

lemma eq_height_0[simp]: "height t = 0 ⟷ t = Leaf"
by(cases t) auto

lemma eq_0_height[simp]: "0 = height t ⟷ 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 ≤ size (t::'a tree)"
by (induction t) auto

lemma size1_height: "size1 t ≤ 2 ^ height (t::'a tree)"
proof(induction t)
case (Node l a r)
show ?case
proof (cases "height l ≤ height r")
case True
have "size1(Node l a r) = size1 l + size1 r" by simp
also have "… ≤ 2 ^ height l + 2 ^ height r" using Node.IH by arith
also have "… ≤ 2 ^ height r + 2 ^ height r" using True by simp
also have "… = 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 "… ≤ 2 ^ height l + 2 ^ height r" using Node.IH by arith
also have "… ≤ 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 ≤ 2 ^ height (t::'a tree) - 1"
using size1_height[of t, unfolded size1_size] by(arith)

lemma height_subtrees: "s ∈ subtrees t ⟹ height s ≤ height t"
by (induction t) auto

lemma min_height_le_height: "min_height t ≤ 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 ≤ size1 t"
proof(induction t)
case (Node l a r)
have "(2::nat) ^ min_height (Node l a r) ≤ 2 ^ min_height l + 2 ^ min_height r"
by (simp add: min_def)
also have "… ≤ size1(Node l a r)" using Node.IH by simp
finally show ?case .
qed simp

subsection ‹\<^const>‹complete››

lemma complete_iff_height: "complete t ⟷ (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 ⟹ size1 t = 2 ^ height t"
by (induction t) auto

lemma size_if_complete: "complete t ⟹ size t = 2 ^ height t - 1"
using size1_if_complete[simplified size1_size] by fastforce

lemma size1_height_if_incomplete:
"¬ complete t ⟹ size1 t < 2 ^ height t"
proof(induction t)
case Leaf thus ?case by simp
next
case (Node l x r)
have 1: ?case if h: "height l < height r"
using h size1_height[of l] size1_height[of r] power_strict_increasing[OF h, of "2::nat"]
by(auto simp: max_def simp del: power_strict_increasing_iff)
have 2: ?case if h: "height l > height r"
using h size1_height[of l] size1_height[of r] power_strict_increasing[OF h, of "2::nat"]
by(auto simp: max_def simp del: power_strict_increasing_iff)
have 3: ?case if h: "height l = height r" and c: "¬ complete l"
using h size1_height[of r] Node.IH(1)[OF c] by(simp)
have 4: ?case if h: "height l = height r" and c: "¬ complete r"
using h size1_height[of l] Node.IH(2)[OF c] by(simp)
from 1 2 3 4 Node.prems show ?case apply (simp add: max_def) by linarith
qed

lemma complete_iff_min_height: "complete t ⟷ (height t = min_height t)"
by(auto simp add: complete_iff_height)

lemma min_height_size1_if_incomplete:
"¬ complete t ⟹ 2 ^ min_height t < size1 t"
proof(induction t)
case Leaf thus ?case by simp
next
case (Node l x r)
have 1: ?case if h: "min_height l < min_height r"
using h min_height_size1[of l] min_height_size1[of r] power_strict_increasing[OF h, of "2::nat"]
by(auto simp: max_def simp del: power_strict_increasing_iff)
have 2: ?case if h: "min_height l > min_height r"
using h min_height_size1[of l] min_height_size1[of r] power_strict_increasing[OF h, of "2::nat"]
by(auto simp: max_def simp del: power_strict_increasing_iff)
have 3: ?case if h: "min_height l = min_height r" and c: "¬ complete l"
using h min_height_size1[of r] Node.IH(1)[OF c] by(simp add: complete_iff_min_height)
have 4: ?case if h: "min_height l = min_height r" and c: "¬ complete r"
using h min_height_size1[of l] Node.IH(2)[OF c] by(simp add: complete_iff_min_height)
from 1 2 3 4 Node.prems show ?case
by (fastforce simp: complete_iff_min_height[THEN iffD1])
qed

lemma complete_if_size1_height: "size1 t = 2 ^ height t ⟹ complete t"
using  size1_height_if_incomplete by fastforce

lemma complete_if_size1_min_height: "size1 t = 2 ^ min_height t ⟹ complete t"
using min_height_size1_if_incomplete by fastforce

lemma complete_iff_size1: "complete t ⟷ size1 t = 2 ^ height t"
using complete_if_size1_height size1_if_complete by blast

subsection ‹\<^const>‹acomplete››

lemma acomplete_subtreeL: "acomplete (Node l x r) ⟹ acomplete l"

lemma acomplete_subtreeR: "acomplete (Node l x r) ⟹ acomplete r"

lemma acomplete_subtrees: "⟦ acomplete t; s ∈ subtrees t ⟧ ⟹ acomplete s"
using [[simp_depth_limit=1]]
by(induction t arbitrary: s)
(auto simp add: acomplete_subtreeL acomplete_subtreeR)

text‹Balanced trees have optimal height:›

lemma acomplete_optimal:
fixes t :: "'a tree" and t' :: "'b tree"
assumes "acomplete t" "size t ≤ size t'" shows "height t ≤ height t'"
proof (cases "complete t")
case True
have "(2::nat) ^ height t ≤ 2 ^ height t'"
proof -
have "2 ^ height t = size1 t"
using True by (simp add: size1_if_complete)
also have "… ≤ size1 t'" using assms(2) by(simp add: size1_size)
also have "… ≤ 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 "… ≤ size1 t'" using assms(2) by (simp add: size1_size)
also have "… ≤ 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 acomplete_def)
with * show ?thesis by arith
qed

subsection ‹\<^const>‹wbalanced››

lemma wbalanced_subtrees: "⟦ wbalanced t; s ∈ subtrees t ⟧ ⟹ wbalanced s"
using [[simp_depth_limit=1]] by(induction t arbitrary: s) auto

subsection ‹\<^const>‹ipl››

text ‹The internal path length of a tree:›

lemma ipl_if_complete_int:
"complete t ⟹ 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 = [] ⟷ t = Leaf"
by (cases t) auto

lemma eq_Nil_inorder[simp]: "[] = inorder t ⟷ 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 ‹Binary Search Tree›

lemma bst_wrt_mono: "(⋀x y. P x y ⟹ Q x y) ⟹ bst_wrt P t ⟹ bst_wrt Q t"
by (induction t) (auto)

lemma bst_wrt_le_if_bst: "bst t ⟹ bst_wrt (≤) t"
using bst_wrt_mono less_imp_le by blast

lemma bst_wrt_le_iff_sorted: "bst_wrt (≤) t ⟷ sorted (inorder t)"
apply (induction t)
apply(simp)
by (fastforce simp: sorted_append intro: less_imp_le less_trans)

lemma bst_iff_sorted_wrt_less: "bst t ⟷ sorted_wrt (<) (inorder t)"
apply (induction t)
apply simp
apply (fastforce simp: sorted_wrt_append)
done

subsection ‹\<^const>‹heap››

subsection ‹\<^const>‹mirror››

lemma mirror_Leaf[simp]: "mirror t = ⟨⟩ ⟷ t = ⟨⟩"
by (induction t) simp_all

lemma Leaf_mirror[simp]: "⟨⟩ = mirror t ⟷ t = ⟨⟩"
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_size)

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
```