1(* Author: Tobias Nipkow *)
2
3section \<open>Creating Balanced Trees\<close>
4
5theory Balance
6imports
7  "HOL-Library.Tree_Real"
8begin
9
10fun bal :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a tree * 'a list" where
11"bal n xs = (if n=0 then (Leaf,xs) else
12 (let m = n div 2;
13      (l, ys) = bal m xs;
14      (r, zs) = bal (n-1-m) (tl ys)
15  in (Node l (hd ys) r, zs)))"
16
17declare bal.simps[simp del]
18
19definition bal_list :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a tree" where
20"bal_list n xs = fst (bal n xs)"
21
22definition balance_list :: "'a list \<Rightarrow> 'a tree" where
23"balance_list xs = bal_list (length xs) xs"
24
25definition bal_tree :: "nat \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
26"bal_tree n t = bal_list n (inorder t)"
27
28definition balance_tree :: "'a tree \<Rightarrow> 'a tree" where
29"balance_tree t = bal_tree (size t) t"
30
31lemma bal_simps:
32  "bal 0 xs = (Leaf, xs)"
33  "n > 0 \<Longrightarrow>
34   bal n xs =
35  (let m = n div 2;
36      (l, ys) = bal m xs;
37      (r, zs) = bal (n-1-m) (tl ys)
38  in (Node l (hd ys) r, zs))"
39by(simp_all add: bal.simps)
40
41text\<open>Some of the following lemmas take advantage of the fact
42that \<open>bal xs n\<close> yields a result even if \<open>n > length xs\<close>.\<close>
43  
44lemma size_bal: "bal n xs = (t,ys) \<Longrightarrow> size t = n"
45proof(induction n xs arbitrary: t ys rule: bal.induct)
46  case (1 n xs)
47  thus ?case
48    by(cases "n=0")
49      (auto simp add: bal_simps Let_def split: prod.splits)
50qed
51
52lemma bal_inorder:
53  "\<lbrakk> bal n xs = (t,ys); n \<le> length xs \<rbrakk>
54  \<Longrightarrow> inorder t = take n xs \<and> ys = drop n xs"
55proof(induction n xs arbitrary: t ys rule: bal.induct)
56  case (1 n xs) show ?case
57  proof cases
58    assume "n = 0" thus ?thesis using 1 by (simp add: bal_simps)
59  next
60    assume [arith]: "n \<noteq> 0"
61    let ?n1 = "n div 2" let ?n2 = "n - 1 - ?n1"
62    from "1.prems" obtain l r xs' where
63      b1: "bal ?n1 xs = (l,xs')" and
64      b2: "bal ?n2 (tl xs') = (r,ys)" and
65      t: "t = \<langle>l, hd xs', r\<rangle>"
66      by(auto simp: Let_def bal_simps split: prod.splits)
67    have IH1: "inorder l = take ?n1 xs \<and> xs' = drop ?n1 xs"
68      using b1 "1.prems" by(intro "1.IH"(1)) auto
69    have IH2: "inorder r = take ?n2 (tl xs') \<and> ys = drop ?n2 (tl xs')"
70      using b1 b2 IH1 "1.prems" by(intro "1.IH"(2)) auto
71    have "drop (n div 2) xs \<noteq> []" using "1.prems"(2) by simp
72    hence "hd (drop ?n1 xs) # take ?n2 (tl (drop ?n1 xs)) = take (?n2 + 1) (drop ?n1 xs)"
73      by (metis Suc_eq_plus1 take_Suc)
74    hence *: "inorder t = take n xs" using t IH1 IH2
75      using take_add[of ?n1 "?n2+1" xs] by(simp)
76    have "n - n div 2 + n div 2 = n" by simp
77    hence "ys = drop n xs" using IH1 IH2 by (simp add: drop_Suc[symmetric])
78    thus ?thesis using * by blast
79  qed
80qed
81
82corollary inorder_bal_list[simp]:
83  "n \<le> length xs \<Longrightarrow> inorder(bal_list n xs) = take n xs"
84unfolding bal_list_def by (metis bal_inorder eq_fst_iff)
85
86corollary inorder_balance_list[simp]: "inorder(balance_list xs) = xs"
87by(simp add: balance_list_def)
88
89corollary inorder_bal_tree:
90  "n \<le> size t \<Longrightarrow> inorder(bal_tree n t) = take n (inorder t)"
91by(simp add: bal_tree_def)
92
93corollary inorder_balance_tree[simp]: "inorder(balance_tree t) = inorder t"
94by(simp add: balance_tree_def inorder_bal_tree)
95
96corollary size_bal_list[simp]: "size(bal_list n xs) = n"
97unfolding bal_list_def by (metis prod.collapse size_bal)
98
99corollary size_balance_list[simp]: "size(balance_list xs) = length xs"
100by (simp add: balance_list_def)
101
102corollary size_bal_tree[simp]: "size(bal_tree n t) = n"
103by(simp add: bal_tree_def)
104
105corollary size_balance_tree[simp]: "size(balance_tree t) = size t"
106by(simp add: balance_tree_def)
107
108lemma min_height_bal:
109  "bal n xs = (t,ys) \<Longrightarrow> min_height t = nat(\<lfloor>log 2 (n + 1)\<rfloor>)"
110proof(induction n xs arbitrary: t ys rule: bal.induct)
111  case (1 n xs) show ?case
112  proof cases
113    assume "n = 0" thus ?thesis
114      using "1.prems" by (simp add: bal_simps)
115  next
116    assume [arith]: "n \<noteq> 0"
117    from "1.prems" obtain l r xs' where
118      b1: "bal (n div 2) xs = (l,xs')" and
119      b2: "bal (n - 1 - n div 2) (tl xs') = (r,ys)" and
120      t: "t = \<langle>l, hd xs', r\<rangle>"
121      by(auto simp: bal_simps Let_def split: prod.splits)
122    let ?log1 = "nat (floor(log 2 (n div 2 + 1)))"
123    let ?log2 = "nat (floor(log 2 (n - 1 - n div 2 + 1)))"
124    have IH1: "min_height l = ?log1" using "1.IH"(1) b1 by simp
125    have IH2: "min_height r = ?log2" using "1.IH"(2) b1 b2 by simp
126    have "(n+1) div 2 \<ge> 1" by arith
127    hence 0: "log 2 ((n+1) div 2) \<ge> 0" by simp
128    have "n - 1 - n div 2 + 1 \<le> n div 2 + 1" by arith
129    hence le: "?log2 \<le> ?log1"
130      by(simp add: nat_mono floor_mono)
131    have "min_height t = min ?log1 ?log2 + 1" by (simp add: t IH1 IH2)
132    also have "\<dots> = ?log2 + 1" using le by (simp add: min_absorb2)
133    also have "n - 1 - n div 2 + 1 = (n+1) div 2" by linarith
134    also have "nat (floor(log 2 ((n+1) div 2))) + 1
135       = nat (floor(log 2 ((n+1) div 2) + 1))"
136      using 0 by linarith
137    also have "\<dots> = nat (floor(log 2 (n + 1)))"
138      using floor_log2_div2[of "n+1"] by (simp add: log_mult)
139    finally show ?thesis .
140  qed
141qed
142
143lemma height_bal:
144  "bal n xs = (t,ys) \<Longrightarrow> height t = nat \<lceil>log 2 (n + 1)\<rceil>"
145proof(induction n xs arbitrary: t ys rule: bal.induct)
146  case (1 n xs) show ?case
147  proof cases
148    assume "n = 0" thus ?thesis
149      using "1.prems" by (simp add: bal_simps)
150  next
151    assume [arith]: "n \<noteq> 0"
152    from "1.prems" obtain l r xs' where
153      b1: "bal (n div 2) xs = (l,xs')" and
154      b2: "bal (n - 1 - n div 2) (tl xs') = (r,ys)" and
155      t: "t = \<langle>l, hd xs', r\<rangle>"
156      by(auto simp: bal_simps Let_def split: prod.splits)
157    let ?log1 = "nat \<lceil>log 2 (n div 2 + 1)\<rceil>"
158    let ?log2 = "nat \<lceil>log 2 (n - 1 - n div 2 + 1)\<rceil>"
159    have IH1: "height l = ?log1" using "1.IH"(1) b1 by simp
160    have IH2: "height r = ?log2" using "1.IH"(2) b1 b2 by simp
161    have 0: "log 2 (n div 2 + 1) \<ge> 0" by auto
162    have "n - 1 - n div 2 + 1 \<le> n div 2 + 1" by arith
163    hence le: "?log2 \<le> ?log1"
164      by(simp add: nat_mono ceiling_mono del: nat_ceiling_le_eq)
165    have "height t = max ?log1 ?log2 + 1" by (simp add: t IH1 IH2)
166    also have "\<dots> = ?log1 + 1" using le by (simp add: max_absorb1)
167    also have "\<dots> = nat \<lceil>log 2 (n div 2 + 1) + 1\<rceil>" using 0 by linarith
168    also have "\<dots> = nat \<lceil>log 2 (n + 1)\<rceil>"
169      using ceiling_log2_div2[of "n+1"] by (simp)
170    finally show ?thesis .
171  qed
172qed
173
174lemma balanced_bal:
175  assumes "bal n xs = (t,ys)" shows "balanced t"
176unfolding balanced_def
177using height_bal[OF assms] min_height_bal[OF assms]
178by linarith
179
180lemma height_bal_list:
181  "n \<le> length xs \<Longrightarrow> height (bal_list n xs) = nat \<lceil>log 2 (n + 1)\<rceil>"
182unfolding bal_list_def by (metis height_bal prod.collapse)
183
184lemma height_balance_list:
185  "height (balance_list xs) = nat \<lceil>log 2 (length xs + 1)\<rceil>"
186by (simp add: balance_list_def height_bal_list)
187
188corollary height_bal_tree:
189  "n \<le> length xs \<Longrightarrow> height (bal_tree n t) = nat\<lceil>log 2 (n + 1)\<rceil>"
190unfolding bal_list_def bal_tree_def
191using height_bal prod.exhaust_sel by blast
192
193corollary height_balance_tree:
194  "height (balance_tree t) = nat\<lceil>log 2 (size t + 1)\<rceil>"
195by (simp add: bal_tree_def balance_tree_def height_bal_list)
196
197corollary balanced_bal_list[simp]: "balanced (bal_list n xs)"
198unfolding bal_list_def by (metis  balanced_bal prod.collapse)
199
200corollary balanced_balance_list[simp]: "balanced (balance_list xs)"
201by (simp add: balance_list_def)
202
203corollary balanced_bal_tree[simp]: "balanced (bal_tree n t)"
204by (simp add: bal_tree_def)
205
206corollary balanced_balance_tree[simp]: "balanced (balance_tree t)"
207by (simp add: balance_tree_def)
208
209lemma wbalanced_bal: "bal n xs = (t,ys) \<Longrightarrow> wbalanced t"
210proof(induction n xs arbitrary: t ys rule: bal.induct)
211  case (1 n xs)
212  show ?case
213  proof cases
214    assume "n = 0"
215    thus ?thesis
216      using "1.prems" by(simp add: bal_simps)
217  next
218    assume "n \<noteq> 0"
219    with "1.prems" obtain l ys r zs where
220      rec1: "bal (n div 2) xs = (l, ys)" and
221      rec2: "bal (n - 1 - n div 2) (tl ys) = (r, zs)" and
222      t: "t = \<langle>l, hd ys, r\<rangle>"
223      by(auto simp add: bal_simps Let_def split: prod.splits)
224    have l: "wbalanced l" using "1.IH"(1)[OF \<open>n\<noteq>0\<close> refl rec1] .
225    have "wbalanced r" using "1.IH"(2)[OF \<open>n\<noteq>0\<close> refl rec1[symmetric] refl rec2] .
226    with l t size_bal[OF rec1] size_bal[OF rec2]
227    show ?thesis by auto
228  qed
229qed
230
231text\<open>An alternative proof via @{thm balanced_if_wbalanced}:\<close>
232lemma "bal n xs = (t,ys) \<Longrightarrow> balanced t"
233by(rule balanced_if_wbalanced[OF wbalanced_bal])
234
235lemma wbalanced_bal_list[simp]: "wbalanced (bal_list n xs)"
236by(simp add: bal_list_def) (metis prod.collapse wbalanced_bal)
237
238lemma wbalanced_balance_list[simp]: "wbalanced (balance_list xs)"
239by(simp add: balance_list_def)
240
241lemma wbalanced_bal_tree[simp]: "wbalanced (bal_tree n t)"
242by(simp add: bal_tree_def)
243
244lemma wbalanced_balance_tree: "wbalanced (balance_tree t)"
245by (simp add: balance_tree_def)
246
247hide_const (open) bal
248
249end
250