src/HOL/Data_Structures/Tree23_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 Tree Implementation of Sets\<close>
```
```     4
```
```     5 theory Tree23_Set
```
```     6 imports
```
```     7   Tree23
```
```     8   Cmp
```
```     9   Set_Specs
```
```    10 begin
```
```    11
```
```    12 declare sorted_wrt.simps(2)[simp del]
```
```    13
```
```    14 definition empty :: "'a tree23" where
```
```    15 "empty = Leaf"
```
```    16
```
```    17 fun isin :: "'a::linorder tree23 \<Rightarrow> 'a \<Rightarrow> bool" where
```
```    18 "isin Leaf x = False" |
```
```    19 "isin (Node2 l a r) x =
```
```    20   (case cmp x a of
```
```    21      LT \<Rightarrow> isin l x |
```
```    22      EQ \<Rightarrow> True |
```
```    23      GT \<Rightarrow> isin r x)" |
```
```    24 "isin (Node3 l a m b r) x =
```
```    25   (case cmp x a of
```
```    26      LT \<Rightarrow> isin l x |
```
```    27      EQ \<Rightarrow> True |
```
```    28      GT \<Rightarrow>
```
```    29        (case cmp x b of
```
```    30           LT \<Rightarrow> isin m x |
```
```    31           EQ \<Rightarrow> True |
```
```    32           GT \<Rightarrow> isin r x))"
```
```    33
```
```    34 datatype 'a up\<^sub>i = T\<^sub>i "'a tree23" | Up\<^sub>i "'a tree23" 'a "'a tree23"
```
```    35
```
```    36 fun tree\<^sub>i :: "'a up\<^sub>i \<Rightarrow> 'a tree23" where
```
```    37 "tree\<^sub>i (T\<^sub>i t) = t" |
```
```    38 "tree\<^sub>i (Up\<^sub>i l a r) = Node2 l a r"
```
```    39
```
```    40 fun ins :: "'a::linorder \<Rightarrow> 'a tree23 \<Rightarrow> 'a up\<^sub>i" where
```
```    41 "ins x Leaf = Up\<^sub>i Leaf x Leaf" |
```
```    42 "ins x (Node2 l a r) =
```
```    43    (case cmp x a of
```
```    44       LT \<Rightarrow>
```
```    45         (case ins x l of
```
```    46            T\<^sub>i l' => T\<^sub>i (Node2 l' a r) |
```
```    47            Up\<^sub>i l1 b l2 => T\<^sub>i (Node3 l1 b l2 a r)) |
```
```    48       EQ \<Rightarrow> T\<^sub>i (Node2 l x r) |
```
```    49       GT \<Rightarrow>
```
```    50         (case ins x r of
```
```    51            T\<^sub>i r' => T\<^sub>i (Node2 l a r') |
```
```    52            Up\<^sub>i r1 b r2 => T\<^sub>i (Node3 l a r1 b r2)))" |
```
```    53 "ins x (Node3 l a m b r) =
```
```    54    (case cmp x a of
```
```    55       LT \<Rightarrow>
```
```    56         (case ins x l of
```
```    57            T\<^sub>i l' => T\<^sub>i (Node3 l' a m b r) |
```
```    58            Up\<^sub>i l1 c l2 => Up\<^sub>i (Node2 l1 c l2) a (Node2 m b r)) |
```
```    59       EQ \<Rightarrow> T\<^sub>i (Node3 l a m b r) |
```
```    60       GT \<Rightarrow>
```
```    61         (case cmp x b of
```
```    62            GT \<Rightarrow>
```
```    63              (case ins x r of
```
```    64                 T\<^sub>i r' => T\<^sub>i (Node3 l a m b r') |
```
```    65                 Up\<^sub>i r1 c r2 => Up\<^sub>i (Node2 l a m) b (Node2 r1 c r2)) |
```
```    66            EQ \<Rightarrow> T\<^sub>i (Node3 l a m b r) |
```
```    67            LT \<Rightarrow>
```
```    68              (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
```
```    72 hide_const insert
```
```    73
```
```    74 definition insert :: "'a::linorder \<Rightarrow> 'a tree23 \<Rightarrow> 'a tree23" where
```
```    75 "insert x t = tree\<^sub>i(ins x t)"
```
```    76
```
```    77 datatype 'a up\<^sub>d = T\<^sub>d "'a tree23" | Up\<^sub>d "'a tree23"
```
```    78
```
```    79 fun tree\<^sub>d :: "'a up\<^sub>d \<Rightarrow> 'a tree23" where
```
```    80 "tree\<^sub>d (T\<^sub>d t) = t" |
```
```    81 "tree\<^sub>d (Up\<^sub>d t) = t"
```
```    82
```
```    83 (* Variation: return None to signal no-change *)
```
```    84
```
```    85 fun node21 :: "'a up\<^sub>d \<Rightarrow> 'a \<Rightarrow> 'a tree23 \<Rightarrow> 'a up\<^sub>d" where
```
```    86 "node21 (T\<^sub>d t1) a t2 = T\<^sub>d(Node2 t1 a t2)" |
```
```    87 "node21 (Up\<^sub>d t1) a (Node2 t2 b t3) = Up\<^sub>d(Node3 t1 a t2 b t3)" |
```
```    88 "node21 (Up\<^sub>d t1) a (Node3 t2 b t3 c t4) = T\<^sub>d(Node2 (Node2 t1 a t2) b (Node2 t3 c t4))"
```
```    89
```
```    90 fun node22 :: "'a tree23 \<Rightarrow> 'a \<Rightarrow> 'a up\<^sub>d \<Rightarrow> 'a up\<^sub>d" where
```
```    91 "node22 t1 a (T\<^sub>d t2) = T\<^sub>d(Node2 t1 a t2)" |
```
```    92 "node22 (Node2 t1 b t2) a (Up\<^sub>d t3) = Up\<^sub>d(Node3 t1 b t2 a t3)" |
```
```    93 "node22 (Node3 t1 b t2 c t3) a (Up\<^sub>d t4) = T\<^sub>d(Node2 (Node2 t1 b t2) c (Node2 t3 a t4))"
```
```    94
```
```    95 fun node31 :: "'a up\<^sub>d \<Rightarrow> 'a \<Rightarrow> 'a tree23 \<Rightarrow> 'a \<Rightarrow> 'a tree23 \<Rightarrow> 'a up\<^sub>d" where
```
```    96 "node31 (T\<^sub>d t1) a t2 b t3 = T\<^sub>d(Node3 t1 a t2 b t3)" |
```
```    97 "node31 (Up\<^sub>d t1) a (Node2 t2 b t3) c t4 = T\<^sub>d(Node2 (Node3 t1 a t2 b t3) c t4)" |
```
```    98 "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)"
```
```    99
```
```   100 fun node32 :: "'a tree23 \<Rightarrow> 'a \<Rightarrow> 'a up\<^sub>d \<Rightarrow> 'a \<Rightarrow> 'a tree23 \<Rightarrow> 'a up\<^sub>d" where
```
```   101 "node32 t1 a (T\<^sub>d t2) b t3 = T\<^sub>d(Node3 t1 a t2 b t3)" |
```
```   102 "node32 t1 a (Up\<^sub>d t2) b (Node2 t3 c t4) = T\<^sub>d(Node2 t1 a (Node3 t2 b t3 c t4))" |
```
```   103 "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))"
```
```   104
```
```   105 fun node33 :: "'a tree23 \<Rightarrow> 'a \<Rightarrow> 'a tree23 \<Rightarrow> 'a \<Rightarrow> 'a up\<^sub>d \<Rightarrow> 'a up\<^sub>d" where
```
```   106 "node33 l a m b (T\<^sub>d r) = T\<^sub>d(Node3 l a m b r)" |
```
```   107 "node33 t1 a (Node2 t2 b t3) c (Up\<^sub>d t4) = T\<^sub>d(Node2 t1 a (Node3 t2 b t3 c t4))" |
```
```   108 "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))"
```
```   109
```
```   110 fun split_min :: "'a tree23 \<Rightarrow> 'a * 'a up\<^sub>d" where
```
```   111 "split_min (Node2 Leaf a Leaf) = (a, Up\<^sub>d Leaf)" |
```
```   112 "split_min (Node3 Leaf a Leaf b Leaf) = (a, T\<^sub>d(Node2 Leaf b Leaf))" |
```
```   113 "split_min (Node2 l a r) = (let (x,l') = split_min l in (x, node21 l' a r))" |
```
```   114 "split_min (Node3 l a m b r) = (let (x,l') = split_min l in (x, node31 l' a m b r))"
```
```   115
```
```   116 text \<open>In the base cases of \<open>split_min\<close> and \<open>del\<close> it is enough to check if one subtree is a \<open>Leaf\<close>,
```
```   117 in which case balancedness implies that so are the others. Exercise.\<close>
```
```   118
```
```   119 fun del :: "'a::linorder \<Rightarrow> 'a tree23 \<Rightarrow> 'a up\<^sub>d" where
```
```   120 "del x Leaf = T\<^sub>d Leaf" |
```
```   121 "del x (Node2 Leaf a Leaf) =
```
```   122   (if x = a then Up\<^sub>d Leaf else T\<^sub>d(Node2 Leaf a Leaf))" |
```
```   123 "del x (Node3 Leaf a Leaf b Leaf) =
```
```   124   T\<^sub>d(if x = a then Node2 Leaf b Leaf else
```
```   125      if x = b then Node2 Leaf a Leaf
```
```   126      else Node3 Leaf a Leaf b Leaf)" |
```
```   127 "del x (Node2 l a r) =
```
```   128   (case cmp x a of
```
```   129      LT \<Rightarrow> node21 (del x l) a r |
```
```   130      GT \<Rightarrow> node22 l a (del x r) |
```
```   131      EQ \<Rightarrow> let (a',t) = split_min r in node22 l a' t)" |
```
```   132 "del x (Node3 l a m b r) =
```
```   133   (case cmp x a of
```
```   134      LT \<Rightarrow> node31 (del x l) a m b r |
```
```   135      EQ \<Rightarrow> let (a',m') = split_min m in node32 l a' m' b r |
```
```   136      GT \<Rightarrow>
```
```   137        (case cmp x b of
```
```   138           LT \<Rightarrow> node32 l a (del x m) b r |
```
```   139           EQ \<Rightarrow> let (b',r') = split_min r in node33 l a m b' r' |
```
```   140           GT \<Rightarrow> node33 l a m b (del x r)))"
```
```   141
```
```   142 definition delete :: "'a::linorder \<Rightarrow> 'a tree23 \<Rightarrow> 'a tree23" where
```
```   143 "delete x t = tree\<^sub>d(del x t)"
```
```   144
```
```   145
```
```   146 subsection "Functional Correctness"
```
```   147
```
```   148 subsubsection "Proofs for isin"
```
```   149
```
```   150 lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> set (inorder t))"
```
```   151 by (induction t) (auto simp: isin_simps ball_Un)
```
```   152
```
```   153
```
```   154 subsubsection "Proofs for insert"
```
```   155
```
```   156 lemma inorder_ins:
```
```   157   "sorted(inorder t) \<Longrightarrow> inorder(tree\<^sub>i(ins x t)) = ins_list x (inorder t)"
```
```   158 by(induction t) (auto simp: ins_list_simps split: up\<^sub>i.splits)
```
```   159
```
```   160 lemma inorder_insert:
```
```   161   "sorted(inorder t) \<Longrightarrow> inorder(insert a t) = ins_list a (inorder t)"
```
```   162 by(simp add: insert_def inorder_ins)
```
```   163
```
```   164
```
```   165 subsubsection "Proofs for delete"
```
```   166
```
```   167 lemma inorder_node21: "height r > 0 \<Longrightarrow>
```
```   168   inorder (tree\<^sub>d (node21 l' a r)) = inorder (tree\<^sub>d l') @ a # inorder r"
```
```   169 by(induct l' a r rule: node21.induct) auto
```
```   170
```
```   171 lemma inorder_node22: "height l > 0 \<Longrightarrow>
```
```   172   inorder (tree\<^sub>d (node22 l a r')) = inorder l @ a # inorder (tree\<^sub>d r')"
```
```   173 by(induct l a r' rule: node22.induct) auto
```
```   174
```
```   175 lemma inorder_node31: "height m > 0 \<Longrightarrow>
```
```   176   inorder (tree\<^sub>d (node31 l' a m b r)) = inorder (tree\<^sub>d l') @ a # inorder m @ b # inorder r"
```
```   177 by(induct l' a m b r rule: node31.induct) auto
```
```   178
```
```   179 lemma inorder_node32: "height r > 0 \<Longrightarrow>
```
```   180   inorder (tree\<^sub>d (node32 l a m' b r)) = inorder l @ a # inorder (tree\<^sub>d m') @ b # inorder r"
```
```   181 by(induct l a m' b r rule: node32.induct) auto
```
```   182
```
```   183 lemma inorder_node33: "height m > 0 \<Longrightarrow>
```
```   184   inorder (tree\<^sub>d (node33 l a m b r')) = inorder l @ a # inorder m @ b # inorder (tree\<^sub>d r')"
```
```   185 by(induct l a m b r' rule: node33.induct) auto
```
```   186
```
```   187 lemmas inorder_nodes = inorder_node21 inorder_node22
```
```   188   inorder_node31 inorder_node32 inorder_node33
```
```   189
```
```   190 lemma split_minD:
```
```   191   "split_min t = (x,t') \<Longrightarrow> bal t \<Longrightarrow> height t > 0 \<Longrightarrow>
```
```   192   x # inorder(tree\<^sub>d t') = inorder t"
```
```   193 by(induction t arbitrary: t' rule: split_min.induct)
```
```   194   (auto simp: inorder_nodes split: prod.splits)
```
```   195
```
```   196 lemma inorder_del: "\<lbrakk> bal t ; sorted(inorder t) \<rbrakk> \<Longrightarrow>
```
```   197   inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)"
```
```   198 by(induction t rule: del.induct)
```
```   199   (auto simp: del_list_simps inorder_nodes split_minD split!: if_split prod.splits)
```
```   200
```
```   201 lemma inorder_delete: "\<lbrakk> bal t ; sorted(inorder t) \<rbrakk> \<Longrightarrow>
```
```   202   inorder(delete x t) = del_list x (inorder t)"
```
```   203 by(simp add: delete_def inorder_del)
```
```   204
```
```   205
```
```   206 subsection \<open>Balancedness\<close>
```
```   207
```
```   208
```
```   209 subsubsection "Proofs for insert"
```
```   210
```
```   211 text\<open>First a standard proof that @{const ins} preserves @{const bal}.\<close>
```
```   212
```
```   213 instantiation up\<^sub>i :: (type)height
```
```   214 begin
```
```   215
```
```   216 fun height_up\<^sub>i :: "'a up\<^sub>i \<Rightarrow> nat" where
```
```   217 "height (T\<^sub>i t) = height t" |
```
```   218 "height (Up\<^sub>i l a r) = height l"
```
```   219
```
```   220 instance ..
```
```   221
```
```   222 end
```
```   223
```
```   224 lemma bal_ins: "bal t \<Longrightarrow> bal (tree\<^sub>i(ins a t)) \<and> height(ins a t) = height t"
```
```   225 by (induct t) (auto split!: if_split up\<^sub>i.split) (* 15 secs in 2015 *)
```
```   226
```
```   227 text\<open>Now an alternative proof (by Brian Huffman) that runs faster because
```
```   228 two properties (balance and height) are combined in one predicate.\<close>
```
```   229
```
```   230 inductive full :: "nat \<Rightarrow> 'a tree23 \<Rightarrow> bool" where
```
```   231 "full 0 Leaf" |
```
```   232 "\<lbrakk>full n l; full n r\<rbrakk> \<Longrightarrow> full (Suc n) (Node2 l p r)" |
```
```   233 "\<lbrakk>full n l; full n m; full n r\<rbrakk> \<Longrightarrow> full (Suc n) (Node3 l p m q r)"
```
```   234
```
```   235 inductive_cases full_elims:
```
```   236   "full n Leaf"
```
```   237   "full n (Node2 l p r)"
```
```   238   "full n (Node3 l p m q r)"
```
```   239
```
```   240 inductive_cases full_0_elim: "full 0 t"
```
```   241 inductive_cases full_Suc_elim: "full (Suc n) t"
```
```   242
```
```   243 lemma full_0_iff [simp]: "full 0 t \<longleftrightarrow> t = Leaf"
```
```   244   by (auto elim: full_0_elim intro: full.intros)
```
```   245
```
```   246 lemma full_Leaf_iff [simp]: "full n Leaf \<longleftrightarrow> n = 0"
```
```   247   by (auto elim: full_elims intro: full.intros)
```
```   248
```
```   249 lemma full_Suc_Node2_iff [simp]:
```
```   250   "full (Suc n) (Node2 l p r) \<longleftrightarrow> full n l \<and> full n r"
```
```   251   by (auto elim: full_elims intro: full.intros)
```
```   252
```
```   253 lemma full_Suc_Node3_iff [simp]:
```
```   254   "full (Suc n) (Node3 l p m q r) \<longleftrightarrow> full n l \<and> full n m \<and> full n r"
```
```   255   by (auto elim: full_elims intro: full.intros)
```
```   256
```
```   257 lemma full_imp_height: "full n t \<Longrightarrow> height t = n"
```
```   258   by (induct set: full, simp_all)
```
```   259
```
```   260 lemma full_imp_bal: "full n t \<Longrightarrow> bal t"
```
```   261   by (induct set: full, auto dest: full_imp_height)
```
```   262
```
```   263 lemma bal_imp_full: "bal t \<Longrightarrow> full (height t) t"
```
```   264   by (induct t, simp_all)
```
```   265
```
```   266 lemma bal_iff_full: "bal t \<longleftrightarrow> (\<exists>n. full n t)"
```
```   267   by (auto elim!: bal_imp_full full_imp_bal)
```
```   268
```
```   269 text \<open>The @{const "insert"} function either preserves the height of the
```
```   270 tree, or increases it by one. The constructor returned by the @{term
```
```   271 "insert"} function determines which: A return value of the form @{term
```
```   272 "T\<^sub>i t"} indicates that the height will be the same. A value of the
```
```   273 form @{term "Up\<^sub>i l p r"} indicates an increase in height.\<close>
```
```   274
```
```   275 fun full\<^sub>i :: "nat \<Rightarrow> 'a up\<^sub>i \<Rightarrow> bool" where
```
```   276 "full\<^sub>i n (T\<^sub>i t) \<longleftrightarrow> full n t" |
```
```   277 "full\<^sub>i n (Up\<^sub>i l p r) \<longleftrightarrow> full n l \<and> full n r"
```
```   278
```
```   279 lemma full\<^sub>i_ins: "full n t \<Longrightarrow> full\<^sub>i n (ins a t)"
```
```   280 by (induct rule: full.induct) (auto split: up\<^sub>i.split)
```
```   281
```
```   282 text \<open>The @{const insert} operation preserves balance.\<close>
```
```   283
```
```   284 lemma bal_insert: "bal t \<Longrightarrow> bal (insert a t)"
```
```   285 unfolding bal_iff_full insert_def
```
```   286 apply (erule exE)
```
```   287 apply (drule full\<^sub>i_ins [of _ _ a])
```
```   288 apply (cases "ins a t")
```
```   289 apply (auto intro: full.intros)
```
```   290 done
```
```   291
```
```   292
```
```   293 subsection "Proofs for delete"
```
```   294
```
```   295 instantiation up\<^sub>d :: (type)height
```
```   296 begin
```
```   297
```
```   298 fun height_up\<^sub>d :: "'a up\<^sub>d \<Rightarrow> nat" where
```
```   299 "height (T\<^sub>d t) = height t" |
```
```   300 "height (Up\<^sub>d t) = height t + 1"
```
```   301
```
```   302 instance ..
```
```   303
```
```   304 end
```
```   305
```
```   306 lemma bal_tree\<^sub>d_node21:
```
```   307   "\<lbrakk>bal r; bal (tree\<^sub>d l'); height r = height l' \<rbrakk> \<Longrightarrow> bal (tree\<^sub>d (node21 l' a r))"
```
```   308 by(induct l' a r rule: node21.induct) auto
```
```   309
```
```   310 lemma bal_tree\<^sub>d_node22:
```
```   311   "\<lbrakk>bal(tree\<^sub>d r'); bal l; height r' = height l \<rbrakk> \<Longrightarrow> bal (tree\<^sub>d (node22 l a r'))"
```
```   312 by(induct l a r' rule: node22.induct) auto
```
```   313
```
```   314 lemma bal_tree\<^sub>d_node31:
```
```   315   "\<lbrakk> bal (tree\<^sub>d l'); bal m; bal r; height l' = height r; height m = height r \<rbrakk>
```
```   316   \<Longrightarrow> bal (tree\<^sub>d (node31 l' a m b r))"
```
```   317 by(induct l' a m b r rule: node31.induct) auto
```
```   318
```
```   319 lemma bal_tree\<^sub>d_node32:
```
```   320   "\<lbrakk> bal l; bal (tree\<^sub>d m'); bal r; height l = height r; height m' = height r \<rbrakk>
```
```   321   \<Longrightarrow> bal (tree\<^sub>d (node32 l a m' b r))"
```
```   322 by(induct l a m' b r rule: node32.induct) auto
```
```   323
```
```   324 lemma bal_tree\<^sub>d_node33:
```
```   325   "\<lbrakk> bal l; bal m; bal(tree\<^sub>d r'); height l = height r'; height m = height r' \<rbrakk>
```
```   326   \<Longrightarrow> bal (tree\<^sub>d (node33 l a m b r'))"
```
```   327 by(induct l a m b r' rule: node33.induct) auto
```
```   328
```
```   329 lemmas bals = bal_tree\<^sub>d_node21 bal_tree\<^sub>d_node22
```
```   330   bal_tree\<^sub>d_node31 bal_tree\<^sub>d_node32 bal_tree\<^sub>d_node33
```
```   331
```
```   332 lemma height'_node21:
```
```   333    "height r > 0 \<Longrightarrow> height(node21 l' a r) = max (height l') (height r) + 1"
```
```   334 by(induct l' a r rule: node21.induct)(simp_all)
```
```   335
```
```   336 lemma height'_node22:
```
```   337    "height l > 0 \<Longrightarrow> height(node22 l a r') = max (height l) (height r') + 1"
```
```   338 by(induct l a r' rule: node22.induct)(simp_all)
```
```   339
```
```   340 lemma height'_node31:
```
```   341   "height m > 0 \<Longrightarrow> height(node31 l a m b r) =
```
```   342    max (height l) (max (height m) (height r)) + 1"
```
```   343 by(induct l a m b r rule: node31.induct)(simp_all add: max_def)
```
```   344
```
```   345 lemma height'_node32:
```
```   346   "height r > 0 \<Longrightarrow> height(node32 l a m b r) =
```
```   347    max (height l) (max (height m) (height r)) + 1"
```
```   348 by(induct l a m b r rule: node32.induct)(simp_all add: max_def)
```
```   349
```
```   350 lemma height'_node33:
```
```   351   "height m > 0 \<Longrightarrow> height(node33 l a m b r) =
```
```   352    max (height l) (max (height m) (height r)) + 1"
```
```   353 by(induct l a m b r rule: node33.induct)(simp_all add: max_def)
```
```   354
```
```   355 lemmas heights = height'_node21 height'_node22
```
```   356   height'_node31 height'_node32 height'_node33
```
```   357
```
```   358 lemma height_split_min:
```
```   359   "split_min t = (x, t') \<Longrightarrow> height t > 0 \<Longrightarrow> bal t \<Longrightarrow> height t' = height t"
```
```   360 by(induct t arbitrary: x t' rule: split_min.induct)
```
```   361   (auto simp: heights split: prod.splits)
```
```   362
```
```   363 lemma height_del: "bal t \<Longrightarrow> height(del x t) = height t"
```
```   364 by(induction x t rule: del.induct)
```
```   365   (auto simp: heights max_def height_split_min split: prod.splits)
```
```   366
```
```   367 lemma bal_split_min:
```
```   368   "\<lbrakk> split_min t = (x, t'); bal t; height t > 0 \<rbrakk> \<Longrightarrow> bal (tree\<^sub>d t')"
```
```   369 by(induct t arbitrary: x t' rule: split_min.induct)
```
```   370   (auto simp: heights height_split_min bals split: prod.splits)
```
```   371
```
```   372 lemma bal_tree\<^sub>d_del: "bal t \<Longrightarrow> bal(tree\<^sub>d(del x t))"
```
```   373 by(induction x t rule: del.induct)
```
```   374   (auto simp: bals bal_split_min height_del height_split_min split: prod.splits)
```
```   375
```
```   376 corollary bal_delete: "bal t \<Longrightarrow> bal(delete x t)"
```
```   377 by(simp add: delete_def bal_tree\<^sub>d_del)
```
```   378
```
```   379
```
```   380 subsection \<open>Overall Correctness\<close>
```
```   381
```
```   382 interpretation S: Set_by_Ordered
```
```   383 where empty = empty and isin = isin and insert = insert and delete = delete
```
```   384 and inorder = inorder and inv = bal
```
```   385 proof (standard, goal_cases)
```
```   386   case 2 thus ?case by(simp add: isin_set)
```
```   387 next
```
```   388   case 3 thus ?case by(simp add: inorder_insert)
```
```   389 next
```
```   390   case 4 thus ?case by(simp add: inorder_delete)
```
```   391 next
```
```   392   case 6 thus ?case by(simp add: bal_insert)
```
```   393 next
```
```   394   case 7 thus ?case by(simp add: bal_delete)
```
```   395 qed (simp add: empty_def)+
```
```   396
```
```   397 end
```