src/HOL/Data_Structures/Tree234_Set.thy
 author nipkow Wed Jun 13 15:24:20 2018 +0200 (10 months ago) changeset 68440 6826718f732d parent 68431 b294e095f64c child 69597 ff784d5a5bfb permissions -rw-r--r--
qualify interpretations to avoid clashes
```     1 (* Author: Tobias Nipkow *)
```
```     2
```
```     3 section \<open>2-3-4 Tree Implementation of Sets\<close>
```
```     4
```
```     5 theory Tree234_Set
```
```     6 imports
```
```     7   Tree234
```
```     8   Cmp
```
```     9   Set_Specs
```
```    10 begin
```
```    11
```
```    12 declare sorted_wrt.simps(2)[simp del]
```
```    13
```
```    14 subsection \<open>Set operations on 2-3-4 trees\<close>
```
```    15
```
```    16 definition empty :: "'a tree234" where
```
```    17 "empty = Leaf"
```
```    18
```
```    19 fun isin :: "'a::linorder tree234 \<Rightarrow> 'a \<Rightarrow> bool" where
```
```    20 "isin Leaf x = False" |
```
```    21 "isin (Node2 l a r) x =
```
```    22   (case cmp x a of LT \<Rightarrow> isin l x | EQ \<Rightarrow> True | GT \<Rightarrow> isin r x)" |
```
```    23 "isin (Node3 l a m b r) x =
```
```    24   (case cmp x a of LT \<Rightarrow> isin l x | EQ \<Rightarrow> True | GT \<Rightarrow> (case cmp x b of
```
```    25    LT \<Rightarrow> isin m x | EQ \<Rightarrow> True | GT \<Rightarrow> isin r x))" |
```
```    26 "isin (Node4 t1 a t2 b t3 c t4) x =
```
```    27   (case cmp x b of
```
```    28      LT \<Rightarrow>
```
```    29        (case cmp x a of
```
```    30           LT \<Rightarrow> isin t1 x |
```
```    31           EQ \<Rightarrow> True |
```
```    32           GT \<Rightarrow> isin t2 x) |
```
```    33      EQ \<Rightarrow> True |
```
```    34      GT \<Rightarrow>
```
```    35        (case cmp x c of
```
```    36           LT \<Rightarrow> isin t3 x |
```
```    37           EQ \<Rightarrow> True |
```
```    38           GT \<Rightarrow> isin t4 x))"
```
```    39
```
```    40 datatype 'a up\<^sub>i = T\<^sub>i "'a tree234" | Up\<^sub>i "'a tree234" 'a "'a tree234"
```
```    41
```
```    42 fun tree\<^sub>i :: "'a up\<^sub>i \<Rightarrow> 'a tree234" where
```
```    43 "tree\<^sub>i (T\<^sub>i t) = t" |
```
```    44 "tree\<^sub>i (Up\<^sub>i l a r) = Node2 l a r"
```
```    45
```
```    46 fun ins :: "'a::linorder \<Rightarrow> 'a tree234 \<Rightarrow> 'a up\<^sub>i" where
```
```    47 "ins x Leaf = Up\<^sub>i Leaf x Leaf" |
```
```    48 "ins x (Node2 l a r) =
```
```    49    (case cmp x a of
```
```    50       LT \<Rightarrow> (case ins x l of
```
```    51               T\<^sub>i l' => T\<^sub>i (Node2 l' a r)
```
```    52             | Up\<^sub>i l1 b l2 => T\<^sub>i (Node3 l1 b l2 a r)) |
```
```    53       EQ \<Rightarrow> T\<^sub>i (Node2 l x r) |
```
```    54       GT \<Rightarrow> (case ins x r of
```
```    55               T\<^sub>i r' => T\<^sub>i (Node2 l a r')
```
```    56             | Up\<^sub>i r1 b r2 => T\<^sub>i (Node3 l a r1 b r2)))" |
```
```    57 "ins x (Node3 l a m b r) =
```
```    58    (case cmp x a of
```
```    59       LT \<Rightarrow> (case ins x l of
```
```    60               T\<^sub>i l' => T\<^sub>i (Node3 l' a m b r)
```
```    61             | Up\<^sub>i l1 c l2 => Up\<^sub>i (Node2 l1 c l2) a (Node2 m b r)) |
```
```    62       EQ \<Rightarrow> T\<^sub>i (Node3 l a m b r) |
```
```    63       GT \<Rightarrow> (case cmp x b of
```
```    64                GT \<Rightarrow> (case ins x r of
```
```    65                        T\<^sub>i r' => T\<^sub>i (Node3 l a m b r')
```
```    66                      | Up\<^sub>i r1 c r2 => Up\<^sub>i (Node2 l a m) b (Node2 r1 c r2)) |
```
```    67                EQ \<Rightarrow> T\<^sub>i (Node3 l a m b r) |
```
```    68                LT \<Rightarrow> (case ins x m of
```
```    69                        T\<^sub>i m' => T\<^sub>i (Node3 l a m' b r)
```
```    70                      | Up\<^sub>i m1 c m2 => Up\<^sub>i (Node2 l a m1) c (Node2 m2 b r))))" |
```
```    71 "ins x (Node4 t1 a t2 b t3 c t4) =
```
```    72   (case cmp x b of
```
```    73      LT \<Rightarrow>
```
```    74        (case cmp x a of
```
```    75           LT \<Rightarrow>
```
```    76             (case ins x t1 of
```
```    77                T\<^sub>i t => T\<^sub>i (Node4 t a t2 b t3 c t4) |
```
```    78                Up\<^sub>i l y r => Up\<^sub>i (Node2 l y r) a (Node3 t2 b t3 c t4)) |
```
```    79           EQ \<Rightarrow> T\<^sub>i (Node4 t1 a t2 b t3 c t4) |
```
```    80           GT \<Rightarrow>
```
```    81             (case ins x t2 of
```
```    82                T\<^sub>i t => T\<^sub>i (Node4 t1 a t b t3 c t4) |
```
```    83                Up\<^sub>i l y r => Up\<^sub>i (Node2 t1 a l) y (Node3 r b t3 c t4))) |
```
```    84      EQ \<Rightarrow> T\<^sub>i (Node4 t1 a t2 b t3 c t4) |
```
```    85      GT \<Rightarrow>
```
```    86        (case cmp x c of
```
```    87           LT \<Rightarrow>
```
```    88             (case ins x t3 of
```
```    89               T\<^sub>i t => T\<^sub>i (Node4 t1 a t2 b t c t4) |
```
```    90               Up\<^sub>i l y r => Up\<^sub>i (Node2 t1 a t2) b (Node3 l y r c t4)) |
```
```    91           EQ \<Rightarrow> T\<^sub>i (Node4 t1 a t2 b t3 c t4) |
```
```    92           GT \<Rightarrow>
```
```    93             (case ins x t4 of
```
```    94               T\<^sub>i t => T\<^sub>i (Node4 t1 a t2 b t3 c t) |
```
```    95               Up\<^sub>i l y r => Up\<^sub>i (Node2 t1 a t2) b (Node3 t3 c l y r))))"
```
```    96
```
```    97 hide_const insert
```
```    98
```
```    99 definition insert :: "'a::linorder \<Rightarrow> 'a tree234 \<Rightarrow> 'a tree234" where
```
```   100 "insert x t = tree\<^sub>i(ins x t)"
```
```   101
```
```   102 datatype 'a up\<^sub>d = T\<^sub>d "'a tree234" | Up\<^sub>d "'a tree234"
```
```   103
```
```   104 fun tree\<^sub>d :: "'a up\<^sub>d \<Rightarrow> 'a tree234" where
```
```   105 "tree\<^sub>d (T\<^sub>d t) = t" |
```
```   106 "tree\<^sub>d (Up\<^sub>d t) = t"
```
```   107
```
```   108 fun node21 :: "'a up\<^sub>d \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a up\<^sub>d" where
```
```   109 "node21 (T\<^sub>d l) a r = T\<^sub>d(Node2 l a r)" |
```
```   110 "node21 (Up\<^sub>d l) a (Node2 lr b rr) = Up\<^sub>d(Node3 l a lr b rr)" |
```
```   111 "node21 (Up\<^sub>d l) a (Node3 lr b mr c rr) = T\<^sub>d(Node2 (Node2 l a lr) b (Node2 mr c rr))" |
```
```   112 "node21 (Up\<^sub>d t1) a (Node4 t2 b t3 c t4 d t5) = T\<^sub>d(Node2 (Node2 t1 a t2) b (Node3 t3 c t4 d t5))"
```
```   113
```
```   114 fun node22 :: "'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a up\<^sub>d \<Rightarrow> 'a up\<^sub>d" where
```
```   115 "node22 l a (T\<^sub>d r) = T\<^sub>d(Node2 l a r)" |
```
```   116 "node22 (Node2 ll b rl) a (Up\<^sub>d r) = Up\<^sub>d(Node3 ll b rl a r)" |
```
```   117 "node22 (Node3 ll b ml c rl) a (Up\<^sub>d r) = T\<^sub>d(Node2 (Node2 ll b ml) c (Node2 rl a r))" |
```
```   118 "node22 (Node4 t1 a t2 b t3 c t4) d (Up\<^sub>d t5) = T\<^sub>d(Node2 (Node2 t1 a t2) b (Node3 t3 c t4 d t5))"
```
```   119
```
```   120 fun node31 :: "'a up\<^sub>d \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a up\<^sub>d" where
```
```   121 "node31 (T\<^sub>d t1) a t2 b t3 = T\<^sub>d(Node3 t1 a t2 b t3)" |
```
```   122 "node31 (Up\<^sub>d t1) a (Node2 t2 b t3) c t4 = T\<^sub>d(Node2 (Node3 t1 a t2 b t3) c t4)" |
```
```   123 "node31 (Up\<^sub>d t1) a (Node3 t2 b t3 c t4) d t5 = T\<^sub>d(Node3 (Node2 t1 a t2) b (Node2 t3 c t4) d t5)" |
```
```   124 "node31 (Up\<^sub>d t1) a (Node4 t2 b t3 c t4 d t5) e t6 = T\<^sub>d(Node3 (Node2 t1 a t2) b (Node3 t3 c t4 d t5) e t6)"
```
```   125
```
```   126 fun node32 :: "'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a up\<^sub>d \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a up\<^sub>d" where
```
```   127 "node32 t1 a (T\<^sub>d t2) b t3 = T\<^sub>d(Node3 t1 a t2 b t3)" |
```
```   128 "node32 t1 a (Up\<^sub>d t2) b (Node2 t3 c t4) = T\<^sub>d(Node2 t1 a (Node3 t2 b t3 c t4))" |
```
```   129 "node32 t1 a (Up\<^sub>d t2) b (Node3 t3 c t4 d t5) = T\<^sub>d(Node3 t1 a (Node2 t2 b t3) c (Node2 t4 d t5))" |
```
```   130 "node32 t1 a (Up\<^sub>d t2) b (Node4 t3 c t4 d t5 e t6) = T\<^sub>d(Node3 t1 a (Node2 t2 b t3) c (Node3 t4 d t5 e t6))"
```
```   131
```
```   132 fun node33 :: "'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a up\<^sub>d \<Rightarrow> 'a up\<^sub>d" where
```
```   133 "node33 l a m b (T\<^sub>d r) = T\<^sub>d(Node3 l a m b r)" |
```
```   134 "node33 t1 a (Node2 t2 b t3) c (Up\<^sub>d t4) = T\<^sub>d(Node2 t1 a (Node3 t2 b t3 c t4))" |
```
```   135 "node33 t1 a (Node3 t2 b t3 c t4) d (Up\<^sub>d t5) = T\<^sub>d(Node3 t1 a (Node2 t2 b t3) c (Node2 t4 d t5))" |
```
```   136 "node33 t1 a (Node4 t2 b t3 c t4 d t5) e (Up\<^sub>d t6) = T\<^sub>d(Node3 t1 a (Node2 t2 b t3) c (Node3 t4 d t5 e t6))"
```
```   137
```
```   138 fun node41 :: "'a up\<^sub>d \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a up\<^sub>d" where
```
```   139 "node41 (T\<^sub>d t1) a t2 b t3 c t4 = T\<^sub>d(Node4 t1 a t2 b t3 c t4)" |
```
```   140 "node41 (Up\<^sub>d t1) a (Node2 t2 b t3) c t4 d t5 = T\<^sub>d(Node3 (Node3 t1 a t2 b t3) c t4 d t5)" |
```
```   141 "node41 (Up\<^sub>d t1) a (Node3 t2 b t3 c t4) d t5 e t6 = T\<^sub>d(Node4 (Node2 t1 a t2) b (Node2 t3 c t4) d t5 e t6)" |
```
```   142 "node41 (Up\<^sub>d t1) a (Node4 t2 b t3 c t4 d t5) e t6 f t7 = T\<^sub>d(Node4 (Node2 t1 a t2) b (Node3 t3 c t4 d t5) e t6 f t7)"
```
```   143
```
```   144 fun node42 :: "'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a up\<^sub>d \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a up\<^sub>d" where
```
```   145 "node42 t1 a (T\<^sub>d t2) b t3 c t4 = T\<^sub>d(Node4 t1 a t2 b t3 c t4)" |
```
```   146 "node42 (Node2 t1 a t2) b (Up\<^sub>d t3) c t4 d t5 = T\<^sub>d(Node3 (Node3 t1 a t2 b t3) c t4 d t5)" |
```
```   147 "node42 (Node3 t1 a t2 b t3) c (Up\<^sub>d t4) d t5 e t6 = T\<^sub>d(Node4 (Node2 t1 a t2) b (Node2 t3 c t4) d t5 e t6)" |
```
```   148 "node42 (Node4 t1 a t2 b t3 c t4) d (Up\<^sub>d t5) e t6 f t7 = T\<^sub>d(Node4 (Node2 t1 a t2) b (Node3 t3 c t4 d t5) e t6 f t7)"
```
```   149
```
```   150 fun node43 :: "'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a up\<^sub>d \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a up\<^sub>d" where
```
```   151 "node43 t1 a t2 b (T\<^sub>d t3) c t4 = T\<^sub>d(Node4 t1 a t2 b t3 c t4)" |
```
```   152 "node43 t1 a (Node2 t2 b t3) c (Up\<^sub>d t4) d t5 = T\<^sub>d(Node3 t1 a (Node3 t2 b t3 c t4) d t5)" |
```
```   153 "node43 t1 a (Node3 t2 b t3 c t4) d (Up\<^sub>d t5) e t6 = T\<^sub>d(Node4 t1 a (Node2 t2 b t3) c (Node2 t4 d t5) e t6)" |
```
```   154 "node43 t1 a (Node4 t2 b t3 c t4 d t5) e (Up\<^sub>d t6) f t7 = T\<^sub>d(Node4 t1 a (Node2 t2 b t3) c (Node3 t4 d t5 e t6) f t7)"
```
```   155
```
```   156 fun node44 :: "'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a tree234 \<Rightarrow> 'a \<Rightarrow> 'a up\<^sub>d \<Rightarrow> 'a up\<^sub>d" where
```
```   157 "node44 t1 a t2 b t3 c (T\<^sub>d t4) = T\<^sub>d(Node4 t1 a t2 b t3 c t4)" |
```
```   158 "node44 t1 a t2 b (Node2 t3 c t4) d (Up\<^sub>d t5) = T\<^sub>d(Node3 t1 a t2 b (Node3 t3 c t4 d t5))" |
```
```   159 "node44 t1 a t2 b (Node3 t3 c t4 d t5) e (Up\<^sub>d t6) = T\<^sub>d(Node4 t1 a t2 b (Node2 t3 c t4) d (Node2 t5 e t6))" |
```
```   160 "node44 t1 a t2 b (Node4 t3 c t4 d t5 e t6) f (Up\<^sub>d t7) = T\<^sub>d(Node4 t1 a t2 b (Node2 t3 c t4) d (Node3 t5 e t6 f t7))"
```
```   161
```
```   162 fun split_min :: "'a tree234 \<Rightarrow> 'a * 'a up\<^sub>d" where
```
```   163 "split_min (Node2 Leaf a Leaf) = (a, Up\<^sub>d Leaf)" |
```
```   164 "split_min (Node3 Leaf a Leaf b Leaf) = (a, T\<^sub>d(Node2 Leaf b Leaf))" |
```
```   165 "split_min (Node4 Leaf a Leaf b Leaf c Leaf) = (a, T\<^sub>d(Node3 Leaf b Leaf c Leaf))" |
```
```   166 "split_min (Node2 l a r) = (let (x,l') = split_min l in (x, node21 l' a r))" |
```
```   167 "split_min (Node3 l a m b r) = (let (x,l') = split_min l in (x, node31 l' a m b r))" |
```
```   168 "split_min (Node4 l a m b n c r) = (let (x,l') = split_min l in (x, node41 l' a m b n c r))"
```
```   169
```
```   170 fun del :: "'a::linorder \<Rightarrow> 'a tree234 \<Rightarrow> 'a up\<^sub>d" where
```
```   171 "del k Leaf = T\<^sub>d Leaf" |
```
```   172 "del k (Node2 Leaf p Leaf) = (if k=p then Up\<^sub>d Leaf else T\<^sub>d(Node2 Leaf p Leaf))" |
```
```   173 "del k (Node3 Leaf p Leaf q Leaf) = T\<^sub>d(if k=p then Node2 Leaf q Leaf
```
```   174   else if k=q then Node2 Leaf p Leaf else Node3 Leaf p Leaf q Leaf)" |
```
```   175 "del k (Node4 Leaf a Leaf b Leaf c Leaf) =
```
```   176   T\<^sub>d(if k=a then Node3 Leaf b Leaf c Leaf else
```
```   177      if k=b then Node3 Leaf a Leaf c Leaf else
```
```   178      if k=c then Node3 Leaf a Leaf b Leaf
```
```   179      else Node4 Leaf a Leaf b Leaf c Leaf)" |
```
```   180 "del k (Node2 l a r) = (case cmp k a of
```
```   181   LT \<Rightarrow> node21 (del k l) a r |
```
```   182   GT \<Rightarrow> node22 l a (del k r) |
```
```   183   EQ \<Rightarrow> let (a',t) = split_min r in node22 l a' t)" |
```
```   184 "del k (Node3 l a m b r) = (case cmp k a of
```
```   185   LT \<Rightarrow> node31 (del k l) a m b r |
```
```   186   EQ \<Rightarrow> let (a',m') = split_min m in node32 l a' m' b r |
```
```   187   GT \<Rightarrow> (case cmp k b of
```
```   188            LT \<Rightarrow> node32 l a (del k m) b r |
```
```   189            EQ \<Rightarrow> let (b',r') = split_min r in node33 l a m b' r' |
```
```   190            GT \<Rightarrow> node33 l a m b (del k r)))" |
```
```   191 "del k (Node4 l a m b n c r) = (case cmp k b of
```
```   192   LT \<Rightarrow> (case cmp k a of
```
```   193           LT \<Rightarrow> node41 (del k l) a m b n c r |
```
```   194           EQ \<Rightarrow> let (a',m') = split_min m in node42 l a' m' b n c r |
```
```   195           GT \<Rightarrow> node42 l a (del k m) b n c r) |
```
```   196   EQ \<Rightarrow> let (b',n') = split_min n in node43 l a m b' n' c r |
```
```   197   GT \<Rightarrow> (case cmp k c of
```
```   198            LT \<Rightarrow> node43 l a m b (del k n) c r |
```
```   199            EQ \<Rightarrow> let (c',r') = split_min r in node44 l a m b n c' r' |
```
```   200            GT \<Rightarrow> node44 l a m b n c (del k r)))"
```
```   201
```
```   202 definition delete :: "'a::linorder \<Rightarrow> 'a tree234 \<Rightarrow> 'a tree234" where
```
```   203 "delete x t = tree\<^sub>d(del x t)"
```
```   204
```
```   205
```
```   206 subsection "Functional correctness"
```
```   207
```
```   208 subsubsection \<open>Functional correctness of isin:\<close>
```
```   209
```
```   210 lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> set (inorder t))"
```
```   211 by (induction t) (auto simp: isin_simps ball_Un)
```
```   212
```
```   213
```
```   214 subsubsection \<open>Functional correctness of insert:\<close>
```
```   215
```
```   216 lemma inorder_ins:
```
```   217   "sorted(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(ins x t)) = ins_list x (inorder t)"
```
```   218 by(induction t) (auto, auto simp: ins_list_simps split!: if_splits up\<^sub>i.splits)
```
```   219
```
```   220 lemma inorder_insert:
```
```   221   "sorted(inorder t) \<Longrightarrow> inorder(insert a t) = ins_list a (inorder t)"
```
```   222 by(simp add: insert_def inorder_ins)
```
```   223
```
```   224
```
```   225 subsubsection \<open>Functional correctness of delete\<close>
```
```   226
```
```   227 lemma inorder_node21: "height r > 0 \<Longrightarrow>
```
```   228   inorder (tree\<^sub>d (node21 l' a r)) = inorder (tree\<^sub>d l') @ a # inorder r"
```
```   229 by(induct l' a r rule: node21.induct) auto
```
```   230
```
```   231 lemma inorder_node22: "height l > 0 \<Longrightarrow>
```
```   232   inorder (tree\<^sub>d (node22 l a r')) = inorder l @ a # inorder (tree\<^sub>d r')"
```
```   233 by(induct l a r' rule: node22.induct) auto
```
```   234
```
```   235 lemma inorder_node31: "height m > 0 \<Longrightarrow>
```
```   236   inorder (tree\<^sub>d (node31 l' a m b r)) = inorder (tree\<^sub>d l') @ a # inorder m @ b # inorder r"
```
```   237 by(induct l' a m b r rule: node31.induct) auto
```
```   238
```
```   239 lemma inorder_node32: "height r > 0 \<Longrightarrow>
```
```   240   inorder (tree\<^sub>d (node32 l a m' b r)) = inorder l @ a # inorder (tree\<^sub>d m') @ b # inorder r"
```
```   241 by(induct l a m' b r rule: node32.induct) auto
```
```   242
```
```   243 lemma inorder_node33: "height m > 0 \<Longrightarrow>
```
```   244   inorder (tree\<^sub>d (node33 l a m b r')) = inorder l @ a # inorder m @ b # inorder (tree\<^sub>d r')"
```
```   245 by(induct l a m b r' rule: node33.induct) auto
```
```   246
```
```   247 lemma inorder_node41: "height m > 0 \<Longrightarrow>
```
```   248   inorder (tree\<^sub>d (node41 l' a m b n c r)) = inorder (tree\<^sub>d l') @ a # inorder m @ b # inorder n @ c # inorder r"
```
```   249 by(induct l' a m b n c r rule: node41.induct) auto
```
```   250
```
```   251 lemma inorder_node42: "height l > 0 \<Longrightarrow>
```
```   252   inorder (tree\<^sub>d (node42 l a m b n c r)) = inorder l @ a # inorder (tree\<^sub>d m) @ b # inorder n @ c # inorder r"
```
```   253 by(induct l a m b n c r rule: node42.induct) auto
```
```   254
```
```   255 lemma inorder_node43: "height m > 0 \<Longrightarrow>
```
```   256   inorder (tree\<^sub>d (node43 l a m b n c r)) = inorder l @ a # inorder m @ b # inorder(tree\<^sub>d n) @ c # inorder r"
```
```   257 by(induct l a m b n c r rule: node43.induct) auto
```
```   258
```
```   259 lemma inorder_node44: "height n > 0 \<Longrightarrow>
```
```   260   inorder (tree\<^sub>d (node44 l a m b n c r)) = inorder l @ a # inorder m @ b # inorder n @ c # inorder (tree\<^sub>d r)"
```
```   261 by(induct l a m b n c r rule: node44.induct) auto
```
```   262
```
```   263 lemmas inorder_nodes = inorder_node21 inorder_node22
```
```   264   inorder_node31 inorder_node32 inorder_node33
```
```   265   inorder_node41 inorder_node42 inorder_node43 inorder_node44
```
```   266
```
```   267 lemma split_minD:
```
```   268   "split_min t = (x,t') \<Longrightarrow> bal t \<Longrightarrow> height t > 0 \<Longrightarrow>
```
```   269   x # inorder(tree\<^sub>d t') = inorder t"
```
```   270 by(induction t arbitrary: t' rule: split_min.induct)
```
```   271   (auto simp: inorder_nodes split: prod.splits)
```
```   272
```
```   273 lemma inorder_del: "\<lbrakk> bal t ; sorted(inorder t) \<rbrakk> \<Longrightarrow>
```
```   274   inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)"
```
```   275 by(induction t rule: del.induct)
```
```   276   (auto simp: inorder_nodes del_list_simps split_minD split!: if_split prod.splits)
```
```   277   (* 30 secs (2016) *)
```
```   278
```
```   279 lemma inorder_delete: "\<lbrakk> bal t ; sorted(inorder t) \<rbrakk> \<Longrightarrow>
```
```   280   inorder(delete x t) = del_list x (inorder t)"
```
```   281 by(simp add: delete_def inorder_del)
```
```   282
```
```   283
```
```   284 subsection \<open>Balancedness\<close>
```
```   285
```
```   286 subsubsection "Proofs for insert"
```
```   287
```
```   288 text\<open>First a standard proof that @{const ins} preserves @{const bal}.\<close>
```
```   289
```
```   290 instantiation up\<^sub>i :: (type)height
```
```   291 begin
```
```   292
```
```   293 fun height_up\<^sub>i :: "'a up\<^sub>i \<Rightarrow> nat" where
```
```   294 "height (T\<^sub>i t) = height t" |
```
```   295 "height (Up\<^sub>i l a r) = height l"
```
```   296
```
```   297 instance ..
```
```   298
```
```   299 end
```
```   300
```
```   301 lemma bal_ins: "bal t \<Longrightarrow> bal (tree\<^sub>i(ins a t)) \<and> height(ins a t) = height t"
```
```   302 by (induct t) (auto split!: if_split up\<^sub>i.split)
```
```   303
```
```   304
```
```   305 text\<open>Now an alternative proof (by Brian Huffman) that runs faster because
```
```   306 two properties (balance and height) are combined in one predicate.\<close>
```
```   307
```
```   308 inductive full :: "nat \<Rightarrow> 'a tree234 \<Rightarrow> bool" where
```
```   309 "full 0 Leaf" |
```
```   310 "\<lbrakk>full n l; full n r\<rbrakk> \<Longrightarrow> full (Suc n) (Node2 l p r)" |
```
```   311 "\<lbrakk>full n l; full n m; full n r\<rbrakk> \<Longrightarrow> full (Suc n) (Node3 l p m q r)" |
```
```   312 "\<lbrakk>full n l; full n m; full n m'; full n r\<rbrakk> \<Longrightarrow> full (Suc n) (Node4 l p m q m' q' r)"
```
```   313
```
```   314 inductive_cases full_elims:
```
```   315   "full n Leaf"
```
```   316   "full n (Node2 l p r)"
```
```   317   "full n (Node3 l p m q r)"
```
```   318   "full n (Node4 l p m q m' q' r)"
```
```   319
```
```   320 inductive_cases full_0_elim: "full 0 t"
```
```   321 inductive_cases full_Suc_elim: "full (Suc n) t"
```
```   322
```
```   323 lemma full_0_iff [simp]: "full 0 t \<longleftrightarrow> t = Leaf"
```
```   324   by (auto elim: full_0_elim intro: full.intros)
```
```   325
```
```   326 lemma full_Leaf_iff [simp]: "full n Leaf \<longleftrightarrow> n = 0"
```
```   327   by (auto elim: full_elims intro: full.intros)
```
```   328
```
```   329 lemma full_Suc_Node2_iff [simp]:
```
```   330   "full (Suc n) (Node2 l p r) \<longleftrightarrow> full n l \<and> full n r"
```
```   331   by (auto elim: full_elims intro: full.intros)
```
```   332
```
```   333 lemma full_Suc_Node3_iff [simp]:
```
```   334   "full (Suc n) (Node3 l p m q r) \<longleftrightarrow> full n l \<and> full n m \<and> full n r"
```
```   335   by (auto elim: full_elims intro: full.intros)
```
```   336
```
```   337 lemma full_Suc_Node4_iff [simp]:
```
```   338   "full (Suc n) (Node4 l p m q m' q' r) \<longleftrightarrow> full n l \<and> full n m \<and> full n m' \<and> full n r"
```
```   339   by (auto elim: full_elims intro: full.intros)
```
```   340
```
```   341 lemma full_imp_height: "full n t \<Longrightarrow> height t = n"
```
```   342   by (induct set: full, simp_all)
```
```   343
```
```   344 lemma full_imp_bal: "full n t \<Longrightarrow> bal t"
```
```   345   by (induct set: full, auto dest: full_imp_height)
```
```   346
```
```   347 lemma bal_imp_full: "bal t \<Longrightarrow> full (height t) t"
```
```   348   by (induct t, simp_all)
```
```   349
```
```   350 lemma bal_iff_full: "bal t \<longleftrightarrow> (\<exists>n. full n t)"
```
```   351   by (auto elim!: bal_imp_full full_imp_bal)
```
```   352
```
```   353 text \<open>The @{const "insert"} function either preserves the height of the
```
```   354 tree, or increases it by one. The constructor returned by the @{term
```
```   355 "insert"} function determines which: A return value of the form @{term
```
```   356 "T\<^sub>i t"} indicates that the height will be the same. A value of the
```
```   357 form @{term "Up\<^sub>i l p r"} indicates an increase in height.\<close>
```
```   358
```
```   359 primrec full\<^sub>i :: "nat \<Rightarrow> 'a up\<^sub>i \<Rightarrow> bool" where
```
```   360 "full\<^sub>i n (T\<^sub>i t) \<longleftrightarrow> full n t" |
```
```   361 "full\<^sub>i n (Up\<^sub>i l p r) \<longleftrightarrow> full n l \<and> full n r"
```
```   362
```
```   363 lemma full\<^sub>i_ins: "full n t \<Longrightarrow> full\<^sub>i n (ins a t)"
```
```   364 by (induct rule: full.induct) (auto, auto split: up\<^sub>i.split)
```
```   365
```
```   366 text \<open>The @{const insert} operation preserves balance.\<close>
```
```   367
```
```   368 lemma bal_insert: "bal t \<Longrightarrow> bal (insert a t)"
```
```   369 unfolding bal_iff_full insert_def
```
```   370 apply (erule exE)
```
```   371 apply (drule full\<^sub>i_ins [of _ _ a])
```
```   372 apply (cases "ins a t")
```
```   373 apply (auto intro: full.intros)
```
```   374 done
```
```   375
```
```   376
```
```   377 subsubsection "Proofs for delete"
```
```   378
```
```   379 instantiation up\<^sub>d :: (type)height
```
```   380 begin
```
```   381
```
```   382 fun height_up\<^sub>d :: "'a up\<^sub>d \<Rightarrow> nat" where
```
```   383 "height (T\<^sub>d t) = height t" |
```
```   384 "height (Up\<^sub>d t) = height t + 1"
```
```   385
```
```   386 instance ..
```
```   387
```
```   388 end
```
```   389
```
```   390 lemma bal_tree\<^sub>d_node21:
```
```   391   "\<lbrakk>bal r; bal (tree\<^sub>d l); height r = height l \<rbrakk> \<Longrightarrow> bal (tree\<^sub>d (node21 l a r))"
```
```   392 by(induct l a r rule: node21.induct) auto
```
```   393
```
```   394 lemma bal_tree\<^sub>d_node22:
```
```   395   "\<lbrakk>bal(tree\<^sub>d r); bal l; height r = height l \<rbrakk> \<Longrightarrow> bal (tree\<^sub>d (node22 l a r))"
```
```   396 by(induct l a r rule: node22.induct) auto
```
```   397
```
```   398 lemma bal_tree\<^sub>d_node31:
```
```   399   "\<lbrakk> bal (tree\<^sub>d l); bal m; bal r; height l = height r; height m = height r \<rbrakk>
```
```   400   \<Longrightarrow> bal (tree\<^sub>d (node31 l a m b r))"
```
```   401 by(induct l a m b r rule: node31.induct) auto
```
```   402
```
```   403 lemma bal_tree\<^sub>d_node32:
```
```   404   "\<lbrakk> bal l; bal (tree\<^sub>d m); bal r; height l = height r; height m = height r \<rbrakk>
```
```   405   \<Longrightarrow> bal (tree\<^sub>d (node32 l a m b r))"
```
```   406 by(induct l a m b r rule: node32.induct) auto
```
```   407
```
```   408 lemma bal_tree\<^sub>d_node33:
```
```   409   "\<lbrakk> bal l; bal m; bal(tree\<^sub>d r); height l = height r; height m = height r \<rbrakk>
```
```   410   \<Longrightarrow> bal (tree\<^sub>d (node33 l a m b r))"
```
```   411 by(induct l a m b r rule: node33.induct) auto
```
```   412
```
```   413 lemma bal_tree\<^sub>d_node41:
```
```   414   "\<lbrakk> bal (tree\<^sub>d l); bal m; bal n; bal r; height l = height r; height m = height r; height n = height r \<rbrakk>
```
```   415   \<Longrightarrow> bal (tree\<^sub>d (node41 l a m b n c r))"
```
```   416 by(induct l a m b n c r rule: node41.induct) auto
```
```   417
```
```   418 lemma bal_tree\<^sub>d_node42:
```
```   419   "\<lbrakk> bal l; bal (tree\<^sub>d m); bal n; bal r; height l = height r; height m = height r; height n = height r \<rbrakk>
```
```   420   \<Longrightarrow> bal (tree\<^sub>d (node42 l a m b n c r))"
```
```   421 by(induct l a m b n c r rule: node42.induct) auto
```
```   422
```
```   423 lemma bal_tree\<^sub>d_node43:
```
```   424   "\<lbrakk> bal l; bal m; bal (tree\<^sub>d n); bal r; height l = height r; height m = height r; height n = height r \<rbrakk>
```
```   425   \<Longrightarrow> bal (tree\<^sub>d (node43 l a m b n c r))"
```
```   426 by(induct l a m b n c r rule: node43.induct) auto
```
```   427
```
```   428 lemma bal_tree\<^sub>d_node44:
```
```   429   "\<lbrakk> bal l; bal m; bal n; bal (tree\<^sub>d r); height l = height r; height m = height r; height n = height r \<rbrakk>
```
```   430   \<Longrightarrow> bal (tree\<^sub>d (node44 l a m b n c r))"
```
```   431 by(induct l a m b n c r rule: node44.induct) auto
```
```   432
```
```   433 lemmas bals = bal_tree\<^sub>d_node21 bal_tree\<^sub>d_node22
```
```   434   bal_tree\<^sub>d_node31 bal_tree\<^sub>d_node32 bal_tree\<^sub>d_node33
```
```   435   bal_tree\<^sub>d_node41 bal_tree\<^sub>d_node42 bal_tree\<^sub>d_node43 bal_tree\<^sub>d_node44
```
```   436
```
```   437 lemma height_node21:
```
```   438    "height r > 0 \<Longrightarrow> height(node21 l a r) = max (height l) (height r) + 1"
```
```   439 by(induct l a r rule: node21.induct)(simp_all add: max.assoc)
```
```   440
```
```   441 lemma height_node22:
```
```   442    "height l > 0 \<Longrightarrow> height(node22 l a r) = max (height l) (height r) + 1"
```
```   443 by(induct l a r rule: node22.induct)(simp_all add: max.assoc)
```
```   444
```
```   445 lemma height_node31:
```
```   446   "height m > 0 \<Longrightarrow> height(node31 l a m b r) =
```
```   447    max (height l) (max (height m) (height r)) + 1"
```
```   448 by(induct l a m b r rule: node31.induct)(simp_all add: max_def)
```
```   449
```
```   450 lemma height_node32:
```
```   451   "height r > 0 \<Longrightarrow> height(node32 l a m b r) =
```
```   452    max (height l) (max (height m) (height r)) + 1"
```
```   453 by(induct l a m b r rule: node32.induct)(simp_all add: max_def)
```
```   454
```
```   455 lemma height_node33:
```
```   456   "height m > 0 \<Longrightarrow> height(node33 l a m b r) =
```
```   457    max (height l) (max (height m) (height r)) + 1"
```
```   458 by(induct l a m b r rule: node33.induct)(simp_all add: max_def)
```
```   459
```
```   460 lemma height_node41:
```
```   461   "height m > 0 \<Longrightarrow> height(node41 l a m b n c r) =
```
```   462    max (height l) (max (height m) (max (height n) (height r))) + 1"
```
```   463 by(induct l a m b n c r rule: node41.induct)(simp_all add: max_def)
```
```   464
```
```   465 lemma height_node42:
```
```   466   "height l > 0 \<Longrightarrow> height(node42 l a m b n c r) =
```
```   467    max (height l) (max (height m) (max (height n) (height r))) + 1"
```
```   468 by(induct l a m b n c r rule: node42.induct)(simp_all add: max_def)
```
```   469
```
```   470 lemma height_node43:
```
```   471   "height m > 0 \<Longrightarrow> height(node43 l a m b n c r) =
```
```   472    max (height l) (max (height m) (max (height n) (height r))) + 1"
```
```   473 by(induct l a m b n c r rule: node43.induct)(simp_all add: max_def)
```
```   474
```
```   475 lemma height_node44:
```
```   476   "height n > 0 \<Longrightarrow> height(node44 l a m b n c r) =
```
```   477    max (height l) (max (height m) (max (height n) (height r))) + 1"
```
```   478 by(induct l a m b n c r rule: node44.induct)(simp_all add: max_def)
```
```   479
```
```   480 lemmas heights = height_node21 height_node22
```
```   481   height_node31 height_node32 height_node33
```
```   482   height_node41 height_node42 height_node43 height_node44
```
```   483
```
```   484 lemma height_split_min:
```
```   485   "split_min t = (x, t') \<Longrightarrow> height t > 0 \<Longrightarrow> bal t \<Longrightarrow> height t' = height t"
```
```   486 by(induct t arbitrary: x t' rule: split_min.induct)
```
```   487   (auto simp: heights split: prod.splits)
```
```   488
```
```   489 lemma height_del: "bal t \<Longrightarrow> height(del x t) = height t"
```
```   490 by(induction x t rule: del.induct)
```
```   491   (auto simp add: heights height_split_min split!: if_split prod.split)
```
```   492
```
```   493 lemma bal_split_min:
```
```   494   "\<lbrakk> split_min t = (x, t'); bal t; height t > 0 \<rbrakk> \<Longrightarrow> bal (tree\<^sub>d t')"
```
```   495 by(induct t arbitrary: x t' rule: split_min.induct)
```
```   496   (auto simp: heights height_split_min bals split: prod.splits)
```
```   497
```
```   498 lemma bal_tree\<^sub>d_del: "bal t \<Longrightarrow> bal(tree\<^sub>d(del x t))"
```
```   499 by(induction x t rule: del.induct)
```
```   500   (auto simp: bals bal_split_min height_del height_split_min split!: if_split prod.split)
```
```   501
```
```   502 corollary bal_delete: "bal t \<Longrightarrow> bal(delete x t)"
```
```   503 by(simp add: delete_def bal_tree\<^sub>d_del)
```
```   504
```
```   505 subsection \<open>Overall Correctness\<close>
```
```   506
```
```   507 interpretation S: Set_by_Ordered
```
```   508 where empty = empty and isin = isin and insert = insert and delete = delete
```
```   509 and inorder = inorder and inv = bal
```
```   510 proof (standard, goal_cases)
```
```   511   case 2 thus ?case by(simp add: isin_set)
```
```   512 next
```
```   513   case 3 thus ?case by(simp add: inorder_insert)
```
```   514 next
```
```   515   case 4 thus ?case by(simp add: inorder_delete)
```
```   516 next
```
```   517   case 6 thus ?case by(simp add: bal_insert)
```
```   518 next
```
```   519   case 7 thus ?case by(simp add: bal_delete)
```
```   520 qed (simp add: empty_def)+
```
```   521
```
```   522 end
```