src/HOL/Data_Structures/Tree234_Map.thy
 author nipkow Sat Apr 21 08:41:42 2018 +0200 (13 months ago) changeset 68020 6aade817bee5 parent 67965 aaa31cd0caef child 68431 b294e095f64c permissions -rw-r--r--
del_min -> split_min
```     1 (* Author: Tobias Nipkow *)
```
```     2
```
```     3 section \<open>2-3-4 Tree Implementation of Maps\<close>
```
```     4
```
```     5 theory Tree234_Map
```
```     6 imports
```
```     7   Tree234_Set
```
```     8   Map_Specs
```
```     9 begin
```
```    10
```
```    11 subsection \<open>Map operations on 2-3-4 trees\<close>
```
```    12
```
```    13 fun 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
```
```    33 fun 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
```
```    75 definition 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
```
```    78 fun 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
```
```   110 definition delete :: "'a::linorder \<Rightarrow> ('a*'b) tree234 \<Rightarrow> ('a*'b) tree234" where
```
```   111 "delete x t = tree\<^sub>d(del x t)"
```
```   112
```
```   113
```
```   114 subsection "Functional correctness"
```
```   115
```
```   116 lemma lookup_map_of:
```
```   117   "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
```
```   118 by (induction t) (auto simp: map_of_simps split: option.split)
```
```   119
```
```   120
```
```   121 lemma inorder_upd:
```
```   122   "sorted1(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(upd a b t)) = upd_list a b (inorder t)"
```
```   123 by(induction t)
```
```   124   (auto simp: upd_list_simps, auto simp: upd_list_simps split: up\<^sub>i.splits)
```
```   125
```
```   126 lemma inorder_update:
```
```   127   "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
```
```   128 by(simp add: update_def inorder_upd)
```
```   129
```
```   130 lemma inorder_del: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
```
```   131   inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)"
```
```   132 by(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
```
```   136 lemma inorder_delete: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
```
```   137   inorder(delete x t) = del_list x (inorder t)"
```
```   138 by(simp add: delete_def inorder_del)
```
```   139
```
```   140
```
```   141 subsection \<open>Balancedness\<close>
```
```   142
```
```   143 lemma bal_upd: "bal t \<Longrightarrow> bal (tree\<^sub>i(upd x y t)) \<and> height(upd x y t) = height t"
```
```   144 by (induct t) (auto, auto split!: if_split up\<^sub>i.split) (* 20 secs (2015) *)
```
```   145
```
```   146 lemma bal_update: "bal t \<Longrightarrow> bal (update x y t)"
```
```   147 by (simp add: update_def bal_upd)
```
```   148
```
```   149 lemma height_del: "bal t \<Longrightarrow> height(del x t) = height t"
```
```   150 by(induction x t rule: del.induct)
```
```   151   (auto simp add: heights height_split_min split!: if_split prod.split)
```
```   152
```
```   153 lemma bal_tree\<^sub>d_del: "bal t \<Longrightarrow> bal(tree\<^sub>d(del x t))"
```
```   154 by(induction x t rule: del.induct)
```
```   155   (auto simp: bals bal_split_min height_del height_split_min split!: if_split prod.split)
```
```   156
```
```   157 corollary bal_delete: "bal t \<Longrightarrow> bal(delete x t)"
```
```   158 by(simp add: delete_def bal_tree\<^sub>d_del)
```
```   159
```
```   160
```
```   161 subsection \<open>Overall Correctness\<close>
```
```   162
```
```   163 interpretation Map_by_Ordered
```
```   164 where empty = Leaf and lookup = lookup and update = update and delete = delete
```
```   165 and inorder = inorder and inv = bal
```
```   166 proof (standard, goal_cases)
```
```   167   case 2 thus ?case by(simp add: lookup_map_of)
```
```   168 next
```
```   169   case 3 thus ?case by(simp add: inorder_update)
```
```   170 next
```
```   171   case 4 thus ?case by(simp add: inorder_delete)
```
```   172 next
```
```   173   case 6 thus ?case by(simp add: bal_update)
```
```   174 next
```
```   175   case 7 thus ?case by(simp add: bal_delete)
```
```   176 qed simp+
```
```   177
```
```   178 end
```