src/HOL/Data_Structures/Tree23_Map.thy
 author nipkow Sat Apr 21 08:41:42 2018 +0200 (14 months ago) changeset 68020 6aade817bee5 parent 67965 aaa31cd0caef child 68431 b294e095f64c permissions -rw-r--r--
del_min -> split_min
```     1 (* Author: Tobias Nipkow *)
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```     2
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```     3 section \<open>2-3 Tree Implementation of Maps\<close>
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```     4
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```     5 theory Tree23_Map
```
```     6 imports
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```     7   Tree23_Set
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```     8   Map_Specs
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```     9 begin
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```    10
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```    11 fun lookup :: "('a::linorder * 'b) tree23 \<Rightarrow> 'a \<Rightarrow> 'b option" where
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```    12 "lookup Leaf x = None" |
```
```    13 "lookup (Node2 l (a,b) r) x = (case cmp x a of
```
```    14   LT \<Rightarrow> lookup l x |
```
```    15   GT \<Rightarrow> lookup r x |
```
```    16   EQ \<Rightarrow> Some b)" |
```
```    17 "lookup (Node3 l (a1,b1) m (a2,b2) r) x = (case cmp x a1 of
```
```    18   LT \<Rightarrow> lookup l x |
```
```    19   EQ \<Rightarrow> Some b1 |
```
```    20   GT \<Rightarrow> (case cmp x a2 of
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```    21           LT \<Rightarrow> lookup m x |
```
```    22           EQ \<Rightarrow> Some b2 |
```
```    23           GT \<Rightarrow> lookup r x))"
```
```    24
```
```    25 fun upd :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>i" where
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```    26 "upd x y Leaf = Up\<^sub>i Leaf (x,y) Leaf" |
```
```    27 "upd x y (Node2 l ab r) = (case cmp x (fst ab) of
```
```    28    LT \<Rightarrow> (case upd x y l of
```
```    29            T\<^sub>i l' => T\<^sub>i (Node2 l' ab r)
```
```    30          | Up\<^sub>i l1 ab' l2 => T\<^sub>i (Node3 l1 ab' l2 ab r)) |
```
```    31    EQ \<Rightarrow> T\<^sub>i (Node2 l (x,y) r) |
```
```    32    GT \<Rightarrow> (case upd x y r of
```
```    33            T\<^sub>i r' => T\<^sub>i (Node2 l ab r')
```
```    34          | Up\<^sub>i r1 ab' r2 => T\<^sub>i (Node3 l ab r1 ab' r2)))" |
```
```    35 "upd x y (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of
```
```    36    LT \<Rightarrow> (case upd x y l of
```
```    37            T\<^sub>i l' => T\<^sub>i (Node3 l' ab1 m ab2 r)
```
```    38          | Up\<^sub>i l1 ab' l2 => Up\<^sub>i (Node2 l1 ab' l2) ab1 (Node2 m ab2 r)) |
```
```    39    EQ \<Rightarrow> T\<^sub>i (Node3 l (x,y) m ab2 r) |
```
```    40    GT \<Rightarrow> (case cmp x (fst ab2) of
```
```    41            LT \<Rightarrow> (case upd x y m of
```
```    42                    T\<^sub>i m' => T\<^sub>i (Node3 l ab1 m' ab2 r)
```
```    43                  | Up\<^sub>i m1 ab' m2 => Up\<^sub>i (Node2 l ab1 m1) ab' (Node2 m2 ab2 r)) |
```
```    44            EQ \<Rightarrow> T\<^sub>i (Node3 l ab1 m (x,y) r) |
```
```    45            GT \<Rightarrow> (case upd x y r of
```
```    46                    T\<^sub>i r' => T\<^sub>i (Node3 l ab1 m ab2 r')
```
```    47                  | Up\<^sub>i r1 ab' r2 => Up\<^sub>i (Node2 l ab1 m) ab2 (Node2 r1 ab' r2))))"
```
```    48
```
```    49 definition update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where
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```    50 "update a b t = tree\<^sub>i(upd a b t)"
```
```    51
```
```    52 fun del :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) up\<^sub>d" where
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```    53 "del x Leaf = T\<^sub>d Leaf" |
```
```    54 "del x (Node2 Leaf ab1 Leaf) = (if x=fst ab1 then Up\<^sub>d Leaf else T\<^sub>d(Node2 Leaf ab1 Leaf))" |
```
```    55 "del x (Node3 Leaf ab1 Leaf ab2 Leaf) = T\<^sub>d(if x=fst ab1 then Node2 Leaf ab2 Leaf
```
```    56   else if x=fst ab2 then Node2 Leaf ab1 Leaf else Node3 Leaf ab1 Leaf ab2 Leaf)" |
```
```    57 "del x (Node2 l ab1 r) = (case cmp x (fst ab1) of
```
```    58   LT \<Rightarrow> node21 (del x l) ab1 r |
```
```    59   GT \<Rightarrow> node22 l ab1 (del x r) |
```
```    60   EQ \<Rightarrow> let (ab1',t) = split_min r in node22 l ab1' t)" |
```
```    61 "del x (Node3 l ab1 m ab2 r) = (case cmp x (fst ab1) of
```
```    62   LT \<Rightarrow> node31 (del x l) ab1 m ab2 r |
```
```    63   EQ \<Rightarrow> let (ab1',m') = split_min m in node32 l ab1' m' ab2 r |
```
```    64   GT \<Rightarrow> (case cmp x (fst ab2) of
```
```    65            LT \<Rightarrow> node32 l ab1 (del x m) ab2 r |
```
```    66            EQ \<Rightarrow> let (ab2',r') = split_min r in node33 l ab1 m ab2' r' |
```
```    67            GT \<Rightarrow> node33 l ab1 m ab2 (del x r)))"
```
```    68
```
```    69 definition delete :: "'a::linorder \<Rightarrow> ('a*'b) tree23 \<Rightarrow> ('a*'b) tree23" where
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```    70 "delete x t = tree\<^sub>d(del x t)"
```
```    71
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```    72
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```    73 subsection \<open>Functional Correctness\<close>
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```    74
```
```    75 lemma lookup_map_of:
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```    76   "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
```
```    77 by (induction t) (auto simp: map_of_simps split: option.split)
```
```    78
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```    79
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```    80 lemma inorder_upd:
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```    81   "sorted1(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(upd x y t)) = upd_list x y (inorder t)"
```
```    82 by(induction t) (auto simp: upd_list_simps split: up\<^sub>i.splits)
```
```    83
```
```    84 corollary inorder_update:
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```    85   "sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
```
```    86 by(simp add: update_def inorder_upd)
```
```    87
```
```    88
```
```    89 lemma inorder_del: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
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```    90   inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)"
```
```    91 by(induction t rule: del.induct)
```
```    92   (auto simp: del_list_simps inorder_nodes split_minD split!: if_split prod.splits)
```
```    93
```
```    94 corollary inorder_delete: "\<lbrakk> bal t ; sorted1(inorder t) \<rbrakk> \<Longrightarrow>
```
```    95   inorder(delete x t) = del_list x (inorder t)"
```
```    96 by(simp add: delete_def inorder_del)
```
```    97
```
```    98
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```    99 subsection \<open>Balancedness\<close>
```
```   100
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```   101 lemma bal_upd: "bal t \<Longrightarrow> bal (tree\<^sub>i(upd x y t)) \<and> height(upd x y t) = height t"
```
```   102 by (induct t) (auto split!: if_split up\<^sub>i.split)(* 16 secs in 2015 *)
```
```   103
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```   104 corollary bal_update: "bal t \<Longrightarrow> bal (update x y t)"
```
```   105 by (simp add: update_def bal_upd)
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```   106
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```   107
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```   108 lemma height_del: "bal t \<Longrightarrow> height(del x t) = height t"
```
```   109 by(induction x t rule: del.induct)
```
```   110   (auto simp add: heights max_def height_split_min split: prod.split)
```
```   111
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```   112 lemma bal_tree\<^sub>d_del: "bal t \<Longrightarrow> bal(tree\<^sub>d(del x t))"
```
```   113 by(induction x t rule: del.induct)
```
```   114   (auto simp: bals bal_split_min height_del height_split_min split: prod.split)
```
```   115
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```   116 corollary bal_delete: "bal t \<Longrightarrow> bal(delete x t)"
```
```   117 by(simp add: delete_def bal_tree\<^sub>d_del)
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```   118
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```   119
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```   120 subsection \<open>Overall Correctness\<close>
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```   121
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```   122 interpretation Map_by_Ordered
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```   123 where empty = Leaf and lookup = lookup and update = update and delete = delete
```
```   124 and inorder = inorder and inv = bal
```
```   125 proof (standard, goal_cases)
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```   126   case 2 thus ?case by(simp add: lookup_map_of)
```
```   127 next
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```   128   case 3 thus ?case by(simp add: inorder_update)
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```   129 next
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```   130   case 4 thus ?case by(simp add: inorder_delete)
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```   131 next
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```   132   case 6 thus ?case by(simp add: bal_update)
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```   133 next
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```   134   case 7 thus ?case by(simp add: bal_delete)
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```   135 qed simp+
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```   136
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```   137 end
```