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 
 (xset_tree l. P x a)  (xset_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 constmap_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 constsize

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 constset_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 constsubtrees

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 constheight and constmin_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 constcomplete

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 constacomplete

lemma acomplete_subtreeL: "acomplete (Node l x r)  acomplete l"
by(simp add: acomplete_def)

lemma acomplete_subtreeR: "acomplete (Node l x r)  acomplete r"
by(simp add: acomplete_def)

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 constwbalanced

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


subsection constipl

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 constheap


subsection constmirror

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