1(* Author: Tobias Nipkow *)
2
3section \<open>2-3-4 Tree Implementation of Maps\<close>
4
5theory Tree234_Map
6imports
7  Tree234_Set
8  Map_Specs
9begin
10
11subsection \<open>Map operations on 2-3-4 trees\<close>
12
13fun lookup :: "('a::linorder * 'b) tree234 \<Rightarrow> 'a \<Rightarrow> 'b option" where
14"lookup Leaf x = None" |
15"lookup (Node2 l (a,b) r) x = (case cmp x a of
16  LT \<Rightarrow> lookup l x |
17  GT \<Rightarrow> lookup r x |
18  EQ \<Rightarrow> Some b)" |
19"lookup (Node3 l (a1,b1) m (a2,b2) r) x = (case cmp x a1 of
20  LT \<Rightarrow> lookup l x |
21  EQ \<Rightarrow> Some b1 |
22  GT \<Rightarrow> (case cmp x a2 of
23          LT \<Rightarrow> lookup m x |
24          EQ \<Rightarrow> Some b2 |
25          GT \<Rightarrow> lookup r x))" |
26"lookup (Node4 t1 (a1,b1) t2 (a2,b2) t3 (a3,b3) t4) x = (case cmp x a2 of
27  LT \<Rightarrow> (case cmp x a1 of
28           LT \<Rightarrow> lookup t1 x | EQ \<Rightarrow> Some b1 | GT \<Rightarrow> lookup t2 x) |
29  EQ \<Rightarrow> Some b2 |
30  GT \<Rightarrow> (case cmp x a3 of
31           LT \<Rightarrow> lookup t3 x | EQ \<Rightarrow> Some b3 | GT \<Rightarrow> lookup t4 x))"
32
33fun upd :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) up\<^sub>i" where
34"upd x y Leaf = Up\<^sub>i Leaf (x,y) Leaf" |
35"upd x y (Node2 l ab r) = (case cmp x (fst ab) of
36   LT \<Rightarrow> (case upd x y l of
37           T\<^sub>i l' => T\<^sub>i (Node2 l' ab r)
38         | Up\<^sub>i l1 ab' l2 => T\<^sub>i (Node3 l1 ab' l2 ab r)) |
39   EQ \<Rightarrow> T\<^sub>i (Node2 l (x,y) r) |
40   GT \<Rightarrow> (case upd x y r of
41           T\<^sub>i r' => T\<^sub>i (Node2 l ab r')
42         | Up\<^sub>i r1 ab' r2 => T\<^sub>i (Node3 l ab r1 ab' r2)))" |
43"upd x y (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of
44   LT \<Rightarrow> (case upd x y l of
45           T\<^sub>i l' => T\<^sub>i (Node3 l' ab1 m ab2 r)
46         | Up\<^sub>i l1 ab' l2 => Up\<^sub>i (Node2 l1 ab' l2) ab1 (Node2 m ab2 r)) |
47   EQ \<Rightarrow> T\<^sub>i (Node3 l (x,y) m ab2 r) |
48   GT \<Rightarrow> (case cmp x (fst ab2) of
49           LT \<Rightarrow> (case upd x y m of
50                   T\<^sub>i m' => T\<^sub>i (Node3 l ab1 m' ab2 r)
51                 | Up\<^sub>i m1 ab' m2 => Up\<^sub>i (Node2 l ab1 m1) ab' (Node2 m2 ab2 r)) |
52           EQ \<Rightarrow> T\<^sub>i (Node3 l ab1 m (x,y) r) |
53           GT \<Rightarrow> (case upd x y r of
54                   T\<^sub>i r' => T\<^sub>i (Node3 l ab1 m ab2 r')
55                 | Up\<^sub>i r1 ab' r2 => Up\<^sub>i (Node2 l ab1 m) ab2 (Node2 r1 ab' r2))))" |
56"upd x y (Node4 t1 ab1 t2 ab2 t3 ab3 t4) = (case cmp x (fst ab2) of
57   LT \<Rightarrow> (case cmp x (fst ab1) of
58            LT \<Rightarrow> (case upd x y t1 of
59                     T\<^sub>i t1' => T\<^sub>i (Node4 t1' ab1 t2 ab2 t3 ab3 t4)
60                  | Up\<^sub>i t11 q t12 => Up\<^sub>i (Node2 t11 q t12) ab1 (Node3 t2 ab2 t3 ab3 t4)) |
61            EQ \<Rightarrow> T\<^sub>i (Node4 t1 (x,y) t2 ab2 t3 ab3 t4) |
62            GT \<Rightarrow> (case upd x y t2 of
63                    T\<^sub>i t2' => T\<^sub>i (Node4 t1 ab1 t2' ab2 t3 ab3 t4)
64                  | Up\<^sub>i t21 q t22 => Up\<^sub>i (Node2 t1 ab1 t21) q (Node3 t22 ab2 t3 ab3 t4))) |
65   EQ \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 (x,y) t3 ab3 t4) |
66   GT \<Rightarrow> (case cmp x (fst ab3) of
67            LT \<Rightarrow> (case upd x y t3 of
68                    T\<^sub>i t3' \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 ab2 t3' ab3 t4)
69                  | Up\<^sub>i t31 q t32 => Up\<^sub>i (Node2 t1 ab1 t2) ab2(*q*) (Node3 t31 q t32 ab3 t4)) |
70            EQ \<Rightarrow> T\<^sub>i (Node4 t1 ab1 t2 ab2 t3 (x,y) t4) |
71            GT \<Rightarrow> (case upd x y t4 of
72                    T\<^sub>i t4' => T\<^sub>i (Node4 t1 ab1 t2 ab2 t3 ab3 t4')
73                  | Up\<^sub>i t41 q t42 => Up\<^sub>i (Node2 t1 ab1 t2) ab2 (Node3 t3 ab3 t41 q t42))))"
74
75definition update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) tree234" where
76"update x y t = tree\<^sub>i(upd x y t)"
77
78fun del :: "'a::linorder \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) up\<^sub>d" where
79"del x Leaf = T\<^sub>d Leaf" |
80"del x (Node2 Leaf ab1 Leaf) = (if x=fst ab1 then Up\<^sub>d Leaf else T\<^sub>d(Node2 Leaf ab1 Leaf))" |
81"del x (Node3 Leaf ab1 Leaf ab2 Leaf) = T\<^sub>d(if x=fst ab1 then Node2 Leaf ab2 Leaf
82  else if x=fst ab2 then Node2 Leaf ab1 Leaf else Node3 Leaf ab1 Leaf ab2 Leaf)" |
83"del x (Node4 Leaf ab1 Leaf ab2 Leaf ab3 Leaf) =
84  T\<^sub>d(if x = fst ab1 then Node3 Leaf ab2 Leaf ab3 Leaf else
85     if x = fst ab2 then Node3 Leaf ab1 Leaf ab3 Leaf else
86     if x = fst ab3 then Node3 Leaf ab1 Leaf ab2 Leaf
87     else Node4 Leaf ab1 Leaf ab2 Leaf ab3 Leaf)" |
88"del x (Node2 l ab1 r) = (case cmp x (fst ab1) of
89  LT \<Rightarrow> node21 (del x l) ab1 r |
90  GT \<Rightarrow> node22 l ab1 (del x r) |
91  EQ \<Rightarrow> let (ab1',t) = split_min r in node22 l ab1' t)" |
92"del x (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of
93  LT \<Rightarrow> node31 (del x l) ab1 m ab2 r |
94  EQ \<Rightarrow> let (ab1',m') = split_min m in node32 l ab1' m' ab2 r |
95  GT \<Rightarrow> (case cmp x (fst ab2) of
96           LT \<Rightarrow> node32 l ab1 (del x m) ab2 r |
97           EQ \<Rightarrow> let (ab2',r') = split_min r in node33 l ab1 m ab2' r' |
98           GT \<Rightarrow> node33 l ab1 m ab2 (del x r)))" |
99"del x (Node4 t1 ab1 t2 ab2 t3 ab3 t4) = (case cmp x (fst ab2) of
100  LT \<Rightarrow> (case cmp x (fst ab1) of
101           LT \<Rightarrow> node41 (del x t1) ab1 t2 ab2 t3 ab3 t4 |
102           EQ \<Rightarrow> let (ab',t2') = split_min t2 in node42 t1 ab' t2' ab2 t3 ab3 t4 |
103           GT \<Rightarrow> node42 t1 ab1 (del x t2) ab2 t3 ab3 t4) |
104  EQ \<Rightarrow> let (ab',t3') = split_min t3 in node43 t1 ab1 t2 ab' t3' ab3 t4 |
105  GT \<Rightarrow> (case cmp x (fst ab3) of
106          LT \<Rightarrow> node43 t1 ab1 t2 ab2 (del x t3) ab3 t4 |
107          EQ \<Rightarrow> let (ab',t4') = split_min t4 in node44 t1 ab1 t2 ab2 t3 ab' t4' |
108          GT \<Rightarrow> node44 t1 ab1 t2 ab2 t3 ab3 (del x t4)))"
109
110definition delete :: "'a::linorder \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) tree234" where
111"delete x t = tree\<^sub>d(del x t)"
112
113
114subsection "Functional correctness"
115
116lemma lookup_map_of:
117  "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
118by (induction t) (auto simp: map_of_simps split: option.split)
119
120
121lemma inorder_upd:
122  "sorted1(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(upd a b t)) = upd_list a b (inorder t)"
123by(induction t)
124  (auto simp: upd_list_simps, auto simp: upd_list_simps split: up\<^sub>i.splits)
125
126lemma inorder_update:
127  "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
128by(simp add: update_def inorder_upd)
129
130lemma inorder_del: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
131  inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)"
132by(induction t rule: del.induct)
133  (auto simp: del_list_simps inorder_nodes split_minD split!: if_splits prod.splits)
134(* 30 secs (2016) *)
135
136lemma inorder_delete: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
137  inorder(delete x t) = del_list x (inorder t)"
138by(simp add: delete_def inorder_del)
139
140
141subsection \<open>Balancedness\<close>
142
143lemma bal_upd: "bal t \<Longrightarrow> bal (tree\<^sub>i(upd x y t)) \<and> height(upd x y t) = height t"
144by (induct t) (auto, auto split!: if_split up\<^sub>i.split) (* 20 secs (2015) *)
145
146lemma bal_update: "bal t \<Longrightarrow> bal (update x y t)"
147by (simp add: update_def bal_upd)
148
149lemma height_del: "bal t \<Longrightarrow> height(del x t) = height t"
150by(induction x t rule: del.induct)
151  (auto simp add: heights height_split_min split!: if_split prod.split)
152
153lemma bal_tree\<^sub>d_del: "bal t \<Longrightarrow> bal(tree\<^sub>d(del x t))"
154by(induction x t rule: del.induct)
155  (auto simp: bals bal_split_min height_del height_split_min split!: if_split prod.split)
156
157corollary bal_delete: "bal t \<Longrightarrow> bal(delete x t)"
158by(simp add: delete_def bal_tree\<^sub>d_del)
159
160
161subsection \<open>Overall Correctness\<close>
162
163interpretation M: Map_by_Ordered
164where empty = empty and lookup = lookup and update = update and delete = delete
165and inorder = inorder and inv = bal
166proof (standard, goal_cases)
167  case 2 thus ?case by(simp add: lookup_map_of)
168next
169  case 3 thus ?case by(simp add: inorder_update)
170next
171  case 4 thus ?case by(simp add: inorder_delete)
172next
173  case 6 thus ?case by(simp add: bal_update)
174next
175  case 7 thus ?case by(simp add: bal_delete)
176qed (simp add: empty_def)+
177
178end
179