61224
|
1 |
(* Author: Tobias Nipkow *)
|
|
2 |
|
|
3 |
section \<open>Red-Black Tree Implementation of Sets\<close>
|
|
4 |
|
|
5 |
theory RBT_Set
|
|
6 |
imports
|
|
7 |
RBT
|
61581
|
8 |
Cmp
|
61224
|
9 |
Isin2
|
|
10 |
begin
|
|
11 |
|
61749
|
12 |
fun ins :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
|
|
13 |
"ins x Leaf = R Leaf x Leaf" |
|
|
14 |
"ins x (B l a r) =
|
61678
|
15 |
(case cmp x a of
|
61749
|
16 |
LT \<Rightarrow> bal (ins x l) a r |
|
|
17 |
GT \<Rightarrow> bal l a (ins x r) |
|
61678
|
18 |
EQ \<Rightarrow> B l a r)" |
|
61749
|
19 |
"ins x (R l a r) =
|
61678
|
20 |
(case cmp x a of
|
61749
|
21 |
LT \<Rightarrow> R (ins x l) a r |
|
|
22 |
GT \<Rightarrow> R l a (ins x r) |
|
61678
|
23 |
EQ \<Rightarrow> R l a r)"
|
61224
|
24 |
|
61749
|
25 |
definition insert :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
|
|
26 |
"insert x t = paint Black (ins x t)"
|
|
27 |
|
|
28 |
fun del :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
|
|
29 |
and delL :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
|
|
30 |
and delR :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
|
61224
|
31 |
where
|
61749
|
32 |
"del x Leaf = Leaf" |
|
|
33 |
"del x (Node _ l a r) =
|
61678
|
34 |
(case cmp x a of
|
61749
|
35 |
LT \<Rightarrow> delL x l a r |
|
|
36 |
GT \<Rightarrow> delR x l a r |
|
61678
|
37 |
EQ \<Rightarrow> combine l r)" |
|
61749
|
38 |
"delL x (B t1 a t2) b t3 = balL (del x (B t1 a t2)) b t3" |
|
|
39 |
"delL x l a r = R (del x l) a r" |
|
|
40 |
"delR x t1 a (B t2 b t3) = balR t1 a (del x (B t2 b t3))" |
|
|
41 |
"delR x l a r = R l a (del x r)"
|
|
42 |
|
|
43 |
definition delete :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
|
|
44 |
"delete x t = paint Black (del x t)"
|
61224
|
45 |
|
|
46 |
|
|
47 |
subsection "Functional Correctness Proofs"
|
|
48 |
|
61749
|
49 |
lemma inorder_paint: "inorder(paint c t) = inorder t"
|
|
50 |
by(induction t) (auto)
|
|
51 |
|
61224
|
52 |
lemma inorder_bal:
|
|
53 |
"inorder(bal l a r) = inorder l @ a # inorder r"
|
61231
|
54 |
by(induction l a r rule: bal.induct) (auto)
|
61224
|
55 |
|
61749
|
56 |
lemma inorder_ins:
|
|
57 |
"sorted(inorder t) \<Longrightarrow> inorder(ins x t) = ins_list x (inorder t)"
|
|
58 |
by(induction x t rule: ins.induct) (auto simp: ins_list_simps inorder_bal)
|
|
59 |
|
61224
|
60 |
lemma inorder_insert:
|
61749
|
61 |
"sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
|
|
62 |
by (simp add: insert_def inorder_ins inorder_paint)
|
61224
|
63 |
|
|
64 |
lemma inorder_balL:
|
|
65 |
"inorder(balL l a r) = inorder l @ a # inorder r"
|
61749
|
66 |
by(induction l a r rule: balL.induct)(auto simp: inorder_bal inorder_paint)
|
61224
|
67 |
|
|
68 |
lemma inorder_balR:
|
|
69 |
"inorder(balR l a r) = inorder l @ a # inorder r"
|
61749
|
70 |
by(induction l a r rule: balR.induct) (auto simp: inorder_bal inorder_paint)
|
61224
|
71 |
|
|
72 |
lemma inorder_combine:
|
|
73 |
"inorder(combine l r) = inorder l @ inorder r"
|
|
74 |
by(induction l r rule: combine.induct)
|
61231
|
75 |
(auto simp: inorder_balL inorder_balR split: tree.split color.split)
|
61224
|
76 |
|
61749
|
77 |
lemma inorder_del:
|
|
78 |
"sorted(inorder t) \<Longrightarrow> inorder(del x t) = del_list x (inorder t)"
|
|
79 |
"sorted(inorder l) \<Longrightarrow> inorder(delL x l a r) =
|
61678
|
80 |
del_list x (inorder l) @ a # inorder r"
|
61749
|
81 |
"sorted(inorder r) \<Longrightarrow> inorder(delR x l a r) =
|
61224
|
82 |
inorder l @ a # del_list x (inorder r)"
|
61749
|
83 |
by(induction x t and x l a r and x l a r rule: del_delL_delR.induct)
|
61231
|
84 |
(auto simp: del_list_simps inorder_combine inorder_balL inorder_balR)
|
61224
|
85 |
|
61749
|
86 |
lemma inorder_delete:
|
|
87 |
"sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
|
|
88 |
by (auto simp: delete_def inorder_del inorder_paint)
|
|
89 |
|
61581
|
90 |
|
61224
|
91 |
interpretation Set_by_Ordered
|
|
92 |
where empty = Leaf and isin = isin and insert = insert and delete = delete
|
61588
|
93 |
and inorder = inorder and inv = "\<lambda>_. True"
|
61224
|
94 |
proof (standard, goal_cases)
|
|
95 |
case 1 show ?case by simp
|
|
96 |
next
|
|
97 |
case 2 thus ?case by(simp add: isin_set)
|
|
98 |
next
|
|
99 |
case 3 thus ?case by(simp add: inorder_insert)
|
|
100 |
next
|
61749
|
101 |
case 4 thus ?case by(simp add: inorder_delete)
|
|
102 |
qed auto
|
61224
|
103 |
|
61754
|
104 |
|
|
105 |
subsection \<open>Structural invariants\<close>
|
|
106 |
|
|
107 |
fun color :: "'a rbt \<Rightarrow> color" where
|
|
108 |
"color Leaf = Black" |
|
|
109 |
"color (Node c _ _ _) = c"
|
|
110 |
|
|
111 |
fun bheight :: "'a rbt \<Rightarrow> nat" where
|
|
112 |
"bheight Leaf = 0" |
|
|
113 |
"bheight (Node c l x r) = (if c = Black then Suc(bheight l) else bheight l)"
|
|
114 |
|
|
115 |
fun inv1 :: "'a rbt \<Rightarrow> bool" where
|
|
116 |
"inv1 Leaf = True" |
|
|
117 |
"inv1 (Node c l a r) =
|
|
118 |
(inv1 l \<and> inv1 r \<and> (c = Black \<or> color l = Black \<and> color r = Black))"
|
|
119 |
|
|
120 |
fun inv1_root :: "'a rbt \<Rightarrow> bool" \<comment> \<open>Weaker version\<close> where
|
|
121 |
"inv1_root Leaf = True" |
|
|
122 |
"inv1_root (Node c l a r) = (inv1 l \<and> inv1 r)"
|
|
123 |
|
|
124 |
fun inv2 :: "'a rbt \<Rightarrow> bool" where
|
|
125 |
"inv2 Leaf = True" |
|
|
126 |
"inv2 (Node c l x r) = (inv2 l \<and> inv2 r \<and> bheight l = bheight r)"
|
|
127 |
|
|
128 |
lemma inv1_rootI[simp]: "inv1 t \<Longrightarrow> inv1_root t"
|
|
129 |
by (cases t) simp+
|
|
130 |
|
|
131 |
definition rbt :: "'a rbt \<Rightarrow> bool" where
|
|
132 |
"rbt t = (inv1 t \<and> inv2 t \<and> color t = Black)"
|
|
133 |
|
|
134 |
lemma color_paint_Black: "color (paint Black t) = Black"
|
|
135 |
by (cases t) auto
|
|
136 |
|
|
137 |
theorem rbt_Leaf: "rbt Leaf"
|
|
138 |
by (simp add: rbt_def)
|
|
139 |
|
|
140 |
lemma paint_inv1_root: "inv1_root t \<Longrightarrow> inv1_root (paint c t)"
|
|
141 |
by (cases t) auto
|
|
142 |
|
|
143 |
lemma inv1_paint_Black: "inv1_root t \<Longrightarrow> inv1 (paint Black t)"
|
|
144 |
by (cases t) auto
|
|
145 |
|
|
146 |
lemma inv2_paint: "inv2 t \<Longrightarrow> inv2 (paint c t)"
|
|
147 |
by (cases t) auto
|
|
148 |
|
|
149 |
lemma inv1_bal: "\<lbrakk>inv1_root l; inv1_root r\<rbrakk> \<Longrightarrow> inv1 (bal l a r)"
|
|
150 |
by (induct l a r rule: bal.induct) auto
|
|
151 |
|
|
152 |
lemma bheight_bal:
|
|
153 |
"bheight l = bheight r \<Longrightarrow> bheight (bal l a r) = Suc (bheight l)"
|
|
154 |
by (induct l a r rule: bal.induct) auto
|
|
155 |
|
|
156 |
lemma inv2_bal:
|
|
157 |
"\<lbrakk> inv2 l; inv2 r; bheight l = bheight r \<rbrakk> \<Longrightarrow> inv2 (bal l a r)"
|
|
158 |
by (induct l a r rule: bal.induct) auto
|
|
159 |
|
|
160 |
|
|
161 |
subsubsection \<open>Insertion\<close>
|
|
162 |
|
|
163 |
lemma inv1_ins: assumes "inv1 t"
|
|
164 |
shows "color t = Black \<Longrightarrow> inv1 (ins x t)" "inv1_root (ins x t)"
|
|
165 |
using assms
|
|
166 |
by (induct x t rule: ins.induct) (auto simp: inv1_bal)
|
|
167 |
|
|
168 |
lemma inv2_ins: assumes "inv2 t"
|
|
169 |
shows "inv2 (ins x t)" "bheight (ins x t) = bheight t"
|
|
170 |
using assms
|
|
171 |
by (induct x t rule: ins.induct) (auto simp: inv2_bal bheight_bal)
|
|
172 |
|
|
173 |
theorem rbt_ins: "rbt t \<Longrightarrow> rbt (insert x t)"
|
|
174 |
by (simp add: inv1_ins inv2_ins color_paint_Black inv1_paint_Black inv2_paint
|
|
175 |
rbt_def insert_def)
|
|
176 |
|
|
177 |
(*
|
|
178 |
lemma bheight_paintR'[simp]: "color t = Black \<Longrightarrow> bheight (paint Red t) = bheight t - 1"
|
|
179 |
by (cases t) auto
|
|
180 |
|
|
181 |
lemma balL_inv2_with_inv1:
|
|
182 |
assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "inv1 rt"
|
|
183 |
shows "bheight (balL lt a rt) = bheight lt + 1" "inv2 (balL lt a rt)"
|
|
184 |
using assms
|
|
185 |
by (induct lt a rt rule: balL.induct) (auto simp: inv2_bal inv2_paint bheight_bal)
|
|
186 |
|
|
187 |
lemma balL_inv2_app:
|
|
188 |
assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "color rt = Black"
|
|
189 |
shows "inv2 (balL lt a rt)"
|
|
190 |
"bheight (balL lt a rt) = bheight rt"
|
|
191 |
using assms
|
|
192 |
by (induct lt a rt rule: balL.induct) (auto simp add: inv2_bal bheight_bal)
|
|
193 |
|
|
194 |
lemma balL_inv1: "\<lbrakk>inv1_root l; inv1 r; color r = Black\<rbrakk> \<Longrightarrow> inv1 (balL l a r)"
|
|
195 |
by (induct l a r rule: balL.induct) (simp_all add: inv1_bal)
|
|
196 |
|
|
197 |
lemma balL_inv1_root: "\<lbrakk> inv1_root lt; inv1 rt \<rbrakk> \<Longrightarrow> inv1_root (balL lt a rt)"
|
|
198 |
by (induct lt a rt rule: balL.induct) (auto simp: inv1_bal paint_inv1_root)
|
|
199 |
|
|
200 |
lemma balR_inv2_with_inv1:
|
|
201 |
assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt + 1" "inv1 lt"
|
|
202 |
shows "inv2 (balR lt a rt) \<and> bheight (balR lt a rt) = bheight lt"
|
|
203 |
using assms
|
|
204 |
by(induct lt a rt rule: balR.induct)(auto simp: inv2_bal bheight_bal inv2_paint)
|
|
205 |
|
|
206 |
lemma balR_inv1: "\<lbrakk>inv1 a; inv1_root b; color a = Black\<rbrakk> \<Longrightarrow> inv1 (balR a x b)"
|
|
207 |
by (induct a x b rule: balR.induct) (simp_all add: inv1_bal)
|
|
208 |
|
|
209 |
lemma balR_inv1_root: "\<lbrakk> inv1 lt; inv1_root rt \<rbrakk> \<Longrightarrow>inv1_root (balR lt x rt)"
|
|
210 |
by (induct lt x rt rule: balR.induct) (auto simp: inv1_bal paint_inv1_root)
|
|
211 |
|
|
212 |
lemma combine_inv2:
|
|
213 |
assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt"
|
|
214 |
shows "bheight (combine lt rt) = bheight lt" "inv2 (combine lt rt)"
|
|
215 |
using assms
|
|
216 |
by (induct lt rt rule: combine.induct)
|
|
217 |
(auto simp: balL_inv2_app split: tree.splits color.splits)
|
|
218 |
|
|
219 |
lemma combine_inv1:
|
|
220 |
assumes "inv1 lt" "inv1 rt"
|
|
221 |
shows "color lt = Black \<Longrightarrow> color rt = Black \<Longrightarrow> inv1 (combine lt rt)"
|
|
222 |
"inv1_root (combine lt rt)"
|
|
223 |
using assms
|
|
224 |
by (induct lt rt rule: combine.induct)
|
|
225 |
(auto simp: balL_inv1 split: tree.splits color.splits)
|
|
226 |
|
|
227 |
|
|
228 |
lemma
|
|
229 |
assumes "inv2 lt" "inv1 lt"
|
|
230 |
shows
|
|
231 |
"\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
|
|
232 |
inv2 (rbt_del_from_left x lt k v rt) \<and>
|
|
233 |
bheight (rbt_del_from_left x lt k v rt) = bheight lt \<and>
|
|
234 |
(color_of lt = B \<and> color_of rt = B \<and> inv1 (rbt_del_from_left x lt k v rt) \<or>
|
|
235 |
(color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (rbt_del_from_left x lt k v rt))"
|
|
236 |
and "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
|
|
237 |
inv2 (rbt_del_from_right x lt k v rt) \<and>
|
|
238 |
bheight (rbt_del_from_right x lt k v rt) = bheight lt \<and>
|
|
239 |
(color_of lt = B \<and> color_of rt = B \<and> inv1 (rbt_del_from_right x lt k v rt) \<or>
|
|
240 |
(color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (rbt_del_from_right x lt k v rt))"
|
|
241 |
and rbt_del_inv1_inv2: "inv2 (rbt_del x lt) \<and> (color_of lt = R \<and> bheight (rbt_del x lt) = bheight lt \<and> inv1 (rbt_del x lt)
|
|
242 |
\<or> color_of lt = B \<and> bheight (rbt_del x lt) = bheight lt - 1 \<and> inv1l (rbt_del x lt))"
|
|
243 |
using assms
|
|
244 |
proof (induct x lt k v rt and x lt k v rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct)
|
|
245 |
case (2 y c _ y')
|
|
246 |
have "y = y' \<or> y < y' \<or> y > y'" by auto
|
|
247 |
thus ?case proof (elim disjE)
|
|
248 |
assume "y = y'"
|
|
249 |
with 2 show ?thesis by (cases c) (simp add: combine_inv2 combine_inv1)+
|
|
250 |
next
|
|
251 |
assume "y < y'"
|
|
252 |
with 2 show ?thesis by (cases c) auto
|
|
253 |
next
|
|
254 |
assume "y' < y"
|
|
255 |
with 2 show ?thesis by (cases c) auto
|
|
256 |
qed
|
|
257 |
next
|
|
258 |
case (3 y lt z v rta y' ss bb)
|
|
259 |
thus ?case by (cases "color_of (Branch B lt z v rta) = B \<and> color_of bb = B") (simp add: balance_left_inv2_with_inv1 balance_left_inv1 balance_left_inv1l)+
|
|
260 |
next
|
|
261 |
case (5 y a y' ss lt z v rta)
|
|
262 |
thus ?case by (cases "color_of a = B \<and> color_of (Branch B lt z v rta) = B") (simp add: balance_right_inv2_with_inv1 balance_right_inv1 balance_right_inv1l)+
|
|
263 |
next
|
|
264 |
case ("6_1" y a y' ss) thus ?case by (cases "color_of a = B \<and> color_of Empty = B") simp+
|
|
265 |
qed auto
|
|
266 |
|
|
267 |
theorem rbt_delete_is_rbt [simp]: assumes "rbt t" shows "rbt (delete k t)"
|
|
268 |
proof -
|
|
269 |
from assms have "inv2 t" and "inv1 t" unfolding rbt_def by auto
|
|
270 |
hence "inv2 (del k t) \<and> (color t = Red \<and> bheight (del k t) = bheight t \<and> inv1 (del k t) \<or> color t = Black \<and> bheight (del k t) = bheight t - 1 \<and> inv1_root (del k t))"
|
|
271 |
by (rule rbt_del_inv1_inv2)
|
|
272 |
hence "inv2 (del k t) \<and> inv1l (rbt_del k t)" by (cases "color_of t") auto
|
|
273 |
with assms show ?thesis
|
|
274 |
unfolding is_rbt_def rbt_delete_def
|
|
275 |
by (auto intro: paint_rbt_sorted rbt_del_rbt_sorted)
|
|
276 |
qed
|
|
277 |
*)
|
61224
|
278 |
end
|