(* Author: Tobias Nipkow *)
section \<open>Creating Balanced Trees\<close>
theory Balance
imports
"~~/src/HOL/Library/Tree"
"~~/src/HOL/Library/Log_Nat"
begin
fun bal :: "'a list \<Rightarrow> nat \<Rightarrow> 'a tree * 'a list" where
"bal xs n = (if n=0 then (Leaf,xs) else
(let m = n div 2;
(l, ys) = bal xs m;
(r, zs) = bal (tl ys) (n-1-m)
in (Node l (hd ys) r, zs)))"
declare bal.simps[simp del]
definition balance_list :: "'a list \<Rightarrow> 'a tree" where
"balance_list xs = fst (bal xs (length xs))"
definition balance_tree :: "'a tree \<Rightarrow> 'a tree" where
"balance_tree = balance_list o inorder"
lemma bal_simps:
"bal xs 0 = (Leaf, xs)"
"n > 0 \<Longrightarrow>
bal xs n =
(let m = n div 2;
(l, ys) = Balance.bal xs m;
(r, zs) = Balance.bal (tl ys) (n-1-m)
in (Node l (hd ys) r, zs))"
by(simp_all add: bal.simps)
text\<open>The following lemmas take advantage of the fact
that \<open>bal xs n\<close> yields a result even if \<open>n > length xs\<close>.\<close>
lemma size_bal: "bal xs n = (t,ys) \<Longrightarrow> size t = n"
proof(induction xs n arbitrary: t ys rule: bal.induct)
case (1 xs n)
thus ?case
by(cases "n=0")
(auto simp add: bal_simps Let_def split: prod.splits)
qed
lemma bal_inorder:
"\<lbrakk> bal xs n = (t,ys); n \<le> length xs \<rbrakk>
\<Longrightarrow> inorder t = take n xs \<and> ys = drop n xs"
proof(induction xs n arbitrary: t ys rule: bal.induct)
case (1 xs n) show ?case
proof cases
assume "n = 0" thus ?thesis using 1 by (simp add: bal_simps)
next
assume [arith]: "n \<noteq> 0"
let ?n1 = "n div 2" let ?n2 = "n - 1 - ?n1"
from "1.prems" obtain l r xs' where
b1: "bal xs ?n1 = (l,xs')" and
b2: "bal (tl xs') ?n2 = (r,ys)" and
t: "t = \<langle>l, hd xs', r\<rangle>"
by(auto simp: Let_def bal_simps split: prod.splits)
have IH1: "inorder l = take ?n1 xs \<and> xs' = drop ?n1 xs"
using b1 "1.prems" by(intro "1.IH"(1)) auto
have IH2: "inorder r = take ?n2 (tl xs') \<and> ys = drop ?n2 (tl xs')"
using b1 b2 IH1 "1.prems" by(intro "1.IH"(2)) auto
have "drop (n div 2) xs \<noteq> []" using "1.prems"(2) by simp
hence "hd (drop ?n1 xs) # take ?n2 (tl (drop ?n1 xs)) = take (?n2 + 1) (drop ?n1 xs)"
by (metis Suc_eq_plus1 take_Suc)
hence *: "inorder t = take n xs" using t IH1 IH2
using take_add[of ?n1 "?n2+1" xs] by(simp)
have "n - n div 2 + n div 2 = n" by simp
hence "ys = drop n xs" using IH1 IH2 by (simp add: drop_Suc[symmetric])
thus ?thesis using * by blast
qed
qed
corollary inorder_balance_list: "inorder(balance_list xs) = xs"
using bal_inorder[of xs "length xs"]
by (metis balance_list_def order_refl prod.collapse take_all)
lemma bal_height: "bal xs n = (t,ys) \<Longrightarrow> height t = floorlog 2 n"
proof(induction xs n arbitrary: t ys rule: bal.induct)
case (1 xs n) show ?case
proof cases
assume "n = 0" thus ?thesis
using "1.prems" by (simp add: floorlog_def bal_simps)
next
assume [arith]: "n \<noteq> 0"
from "1.prems" obtain l r xs' where
b1: "bal xs (n div 2) = (l,xs')" and
b2: "bal (tl xs') (n - 1 - n div 2) = (r,ys)" and
t: "t = \<langle>l, hd xs', r\<rangle>"
by(auto simp: bal_simps Let_def split: prod.splits)
let ?log1 = "floorlog 2 (n div 2)"
let ?log2 = "floorlog 2 (n - 1 - n div 2)"
have IH1: "height l = ?log1" using "1.IH"(1) b1 by simp
have IH2: "height r = ?log2" using "1.IH"(2) b1 b2 by simp
have "n div 2 \<ge> n - 1 - n div 2" by arith
hence le: "?log2 \<le> ?log1" by(simp add:floorlog_mono)
have "height t = max ?log1 ?log2 + 1" by (simp add: t IH1 IH2)
also have "\<dots> = ?log1 + 1" using le by (simp add: max_absorb1)
also have "\<dots> = floorlog 2 n" by (simp add: compute_floorlog)
finally show ?thesis .
qed
qed
lemma bal_min_height:
"bal xs n = (t,ys) \<Longrightarrow> min_height t = floorlog 2 (n + 1) - 1"
proof(induction xs n arbitrary: t ys rule: bal.induct)
case (1 xs n) show ?case
proof cases
assume "n = 0" thus ?thesis
using "1.prems" by (simp add: floorlog_def bal_simps)
next
assume [arith]: "n \<noteq> 0"
from "1.prems" obtain l r xs' where
b1: "bal xs (n div 2) = (l,xs')" and
b2: "bal (tl xs') (n - 1 - n div 2) = (r,ys)" and
t: "t = \<langle>l, hd xs', r\<rangle>"
by(auto simp: bal_simps Let_def split: prod.splits)
let ?log1 = "floorlog 2 (n div 2 + 1) - 1"
let ?log2 = "floorlog 2 (n - 1 - n div 2 + 1) - 1"
let ?log2' = "floorlog 2 (n - n div 2) - 1"
have "n - 1 - n div 2 + 1 = n - n div 2" by arith
hence IH2: "min_height r = ?log2'" using "1.IH"(2) b1 b2 by simp
have IH1: "min_height l = ?log1" using "1.IH"(1) b1 by simp
have *: "floorlog 2 (n - n div 2) \<ge> 1" by (simp add: floorlog_def)
have "n div 2 + 1 \<ge> n - n div 2" by arith
with * have le: "?log2' \<le> ?log1" by(simp add: floorlog_mono diff_le_mono)
have "min_height t = min ?log1 ?log2' + 1" by (simp add: t IH1 IH2)
also have "\<dots> = ?log2' + 1" using le by (simp add: min_absorb2)
also have "\<dots> = floorlog 2 (n - n div 2)" by(simp add: floorlog_def)
also have "n - n div 2 = (n+1) div 2" by arith
also have "floorlog 2 \<dots> = floorlog 2 (n+1) - 1"
by (simp add: compute_floorlog)
finally show ?thesis .
qed
qed
lemma balanced_bal:
assumes "bal xs n = (t,ys)" shows "balanced t"
proof -
have "floorlog 2 n \<le> floorlog 2 (n+1)" by (rule floorlog_mono) auto
thus ?thesis unfolding balanced_def
using bal_height[OF assms] bal_min_height[OF assms] by linarith
qed
corollary size_balance_list[simp]: "size(balance_list xs) = length xs"
by (metis inorder_balance_list length_inorder)
corollary balanced_balance_list[simp]: "balanced (balance_list xs)"
by (metis balance_list_def balanced_bal prod.collapse)
lemma height_balance_list: "height(balance_list xs) = floorlog 2 (length xs)"
by (metis bal_height balance_list_def prod.collapse)
lemma inorder_balance_tree[simp]: "inorder(balance_tree t) = inorder t"
by(simp add: balance_tree_def inorder_balance_list)
lemma size_balance_tree[simp]: "size(balance_tree t) = size t"
by(simp add: balance_tree_def inorder_balance_list)
corollary balanced_balance_tree[simp]: "balanced (balance_tree t)"
by (simp add: balance_tree_def)
lemma height_balance_tree: "height(balance_tree t) = floorlog 2 (size t)"
by(simp add: balance_tree_def height_balance_list)
lemma wbalanced_bal: "bal xs n = (t,ys) \<Longrightarrow> wbalanced t"
proof(induction xs n arbitrary: t ys rule: bal.induct)
case (1 xs n)
show ?case
proof cases
assume "n = 0"
thus ?thesis
using "1.prems" by(simp add: bal_simps)
next
assume "n \<noteq> 0"
with "1.prems" obtain l ys r zs where
rec1: "bal xs (n div 2) = (l, ys)" and
rec2: "bal (tl ys) (n - 1 - n div 2) = (r, zs)" and
t: "t = \<langle>l, hd ys, r\<rangle>"
by(auto simp add: bal_simps Let_def split: prod.splits)
have l: "wbalanced l" using "1.IH"(1)[OF \<open>n\<noteq>0\<close> refl rec1] .
have "wbalanced r" using "1.IH"(2)[OF \<open>n\<noteq>0\<close> refl rec1[symmetric] refl rec2] .
with l t size_bal[OF rec1] size_bal[OF rec2]
show ?thesis by auto
qed
qed
lemma wbalanced_balance_tree: "wbalanced (balance_tree t)"
by(simp add: balance_tree_def balance_list_def)
(metis prod.collapse wbalanced_bal)
hide_const (open) bal
end