src/HOL/Data_Structures/AVL_Set.thy
 author nipkow Thu Nov 05 08:27:14 2015 +0100 (2015-11-05) changeset 61581 00d9682e8dd7 parent 61428 5e1938107371 child 61588 1d2907d0ed73 permissions -rw-r--r--
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```     1 (*
```
```     2 Author:     Tobias Nipkow
```
```     3 Derived from AFP entry AVL.
```
```     4 *)
```
```     5
```
```     6 section "AVL Tree Implementation of Sets"
```
```     7
```
```     8 theory AVL_Set
```
```     9 imports Cmp Isin2
```
```    10 begin
```
```    11
```
```    12 type_synonym 'a avl_tree = "('a,nat) tree"
```
```    13
```
```    14 text {* Invariant: *}
```
```    15
```
```    16 fun avl :: "'a avl_tree \<Rightarrow> bool" where
```
```    17 "avl Leaf = True" |
```
```    18 "avl (Node h l a r) =
```
```    19  ((height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1) \<and>
```
```    20   h = max (height l) (height r) + 1 \<and> avl l \<and> avl r)"
```
```    21
```
```    22 fun ht :: "'a avl_tree \<Rightarrow> nat" where
```
```    23 "ht Leaf = 0" |
```
```    24 "ht (Node h l a r) = h"
```
```    25
```
```    26 definition node :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    27 "node l a r = Node (max (ht l) (ht r) + 1) l a r"
```
```    28
```
```    29 definition balL :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    30 "balL l a r = (
```
```    31   if ht l = ht r + 2 then (case l of
```
```    32     Node _ bl b br \<Rightarrow> (if ht bl < ht br
```
```    33     then case br of
```
```    34       Node _ cl c cr \<Rightarrow> node (node bl b cl) c (node cr a r)
```
```    35     else node bl b (node br a r)))
```
```    36   else node l a r)"
```
```    37
```
```    38 definition balR :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    39 "balR l a r = (
```
```    40   if ht r = ht l + 2 then (case r of
```
```    41     Node _ bl b br \<Rightarrow> (if ht bl > ht br
```
```    42     then case bl of
```
```    43       Node _ cl c cr \<Rightarrow> node (node l a cl) c (node cr b br)
```
```    44     else node (node l a bl) b br))
```
```    45   else node l a r)"
```
```    46
```
```    47 fun insert :: "'a::cmp \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    48 "insert x Leaf = Node 1 Leaf x Leaf" |
```
```    49 "insert x (Node h l a r) = (case cmp x a of
```
```    50    EQ \<Rightarrow> Node h l a r |
```
```    51    LT \<Rightarrow> balL (insert x l) a r |
```
```    52    GT \<Rightarrow> balR l a (insert x r))"
```
```    53
```
```    54 fun delete_max :: "'a avl_tree \<Rightarrow> 'a avl_tree * 'a" where
```
```    55 "delete_max (Node _ l a Leaf) = (l,a)" |
```
```    56 "delete_max (Node _ l a r) =
```
```    57   (let (r',a') = delete_max r in (balL l a r', a'))"
```
```    58
```
```    59 lemmas delete_max_induct = delete_max.induct[case_names Leaf Node]
```
```    60
```
```    61 fun delete_root :: "'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    62 "delete_root (Node h Leaf a r) = r" |
```
```    63 "delete_root (Node h l a Leaf) = l" |
```
```    64 "delete_root (Node h l a r) =
```
```    65   (let (l', a') = delete_max l in balR l' a' r)"
```
```    66
```
```    67 lemmas delete_root_cases = delete_root.cases[case_names Leaf_t Node_Leaf Node_Node]
```
```    68
```
```    69 fun delete :: "'a::cmp \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    70 "delete _ Leaf = Leaf" |
```
```    71 "delete x (Node h l a r) = (case cmp x a of
```
```    72    EQ \<Rightarrow> delete_root (Node h l a r) |
```
```    73    LT \<Rightarrow> balR (delete x l) a r |
```
```    74    GT \<Rightarrow> balL l a (delete x r))"
```
```    75
```
```    76
```
```    77 subsection {* Functional Correctness Proofs *}
```
```    78
```
```    79 text{* Very different from the AFP/AVL proofs *}
```
```    80
```
```    81
```
```    82 subsubsection "Proofs for insert"
```
```    83
```
```    84 lemma inorder_balL:
```
```    85   "inorder (balL l a r) = inorder l @ a # inorder r"
```
```    86 by (auto simp: node_def balL_def split:tree.splits)
```
```    87
```
```    88 lemma inorder_balR:
```
```    89   "inorder (balR l a r) = inorder l @ a # inorder r"
```
```    90 by (auto simp: node_def balR_def split:tree.splits)
```
```    91
```
```    92 theorem inorder_insert:
```
```    93   "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
```
```    94 by (induct t)
```
```    95    (auto simp: ins_list_simps inorder_balL inorder_balR)
```
```    96
```
```    97
```
```    98 subsubsection "Proofs for delete"
```
```    99
```
```   100 lemma inorder_delete_maxD:
```
```   101   "\<lbrakk> delete_max t = (t',a); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
```
```   102    inorder t' @ [a] = inorder t"
```
```   103 by(induction t arbitrary: t' rule: delete_max.induct)
```
```   104   (auto simp: inorder_balL split: prod.splits tree.split)
```
```   105
```
```   106 lemma inorder_delete_root:
```
```   107   "inorder (delete_root (Node h l a r)) = inorder l @ inorder r"
```
```   108 by(induction "Node h l a r" arbitrary: l a r h rule: delete_root.induct)
```
```   109   (auto simp: inorder_balR inorder_delete_maxD split: prod.splits)
```
```   110
```
```   111 theorem inorder_delete:
```
```   112   "sorted(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
```
```   113 by(induction t)
```
```   114   (auto simp: del_list_simps inorder_balL inorder_balR
```
```   115     inorder_delete_root inorder_delete_maxD split: prod.splits)
```
```   116
```
```   117
```
```   118 subsubsection "Overall functional correctness"
```
```   119
```
```   120 interpretation Set_by_Ordered
```
```   121 where empty = Leaf and isin = isin and insert = insert and delete = delete
```
```   122 and inorder = inorder and wf = "\<lambda>_. True"
```
```   123 proof (standard, goal_cases)
```
```   124   case 1 show ?case by simp
```
```   125 next
```
```   126   case 2 thus ?case by(simp add: isin_set)
```
```   127 next
```
```   128   case 3 thus ?case by(simp add: inorder_insert)
```
```   129 next
```
```   130   case 4 thus ?case by(simp add: inorder_delete)
```
```   131 qed (rule TrueI)+
```
```   132
```
```   133
```
```   134 subsection {* AVL invariants *}
```
```   135
```
```   136 text{* Essentially the AFP/AVL proofs *}
```
```   137
```
```   138
```
```   139 subsubsection {* Insertion maintains AVL balance *}
```
```   140
```
```   141 declare Let_def [simp]
```
```   142
```
```   143 lemma [simp]: "avl t \<Longrightarrow> ht t = height t"
```
```   144 by (induct t) simp_all
```
```   145
```
```   146 lemma height_balL:
```
```   147   "\<lbrakk> height l = height r + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
```
```   148    height (balL l a r) = height r + 2 \<or>
```
```   149    height (balL l a r) = height r + 3"
```
```   150 by (cases l) (auto simp:node_def balL_def split:tree.split)
```
```   151
```
```   152 lemma height_balR:
```
```   153   "\<lbrakk> height r = height l + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
```
```   154    height (balR l a r) = height l + 2 \<or>
```
```   155    height (balR l a r) = height l + 3"
```
```   156 by (cases r) (auto simp add:node_def balR_def split:tree.split)
```
```   157
```
```   158 lemma [simp]: "height(node l a r) = max (height l) (height r) + 1"
```
```   159 by (simp add: node_def)
```
```   160
```
```   161 lemma avl_node:
```
```   162   "\<lbrakk> avl l; avl r;
```
```   163      height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1
```
```   164    \<rbrakk> \<Longrightarrow> avl(node l a r)"
```
```   165 by (auto simp add:max_def node_def)
```
```   166
```
```   167 lemma height_balL2:
```
```   168   "\<lbrakk> avl l; avl r; height l \<noteq> height r + 2 \<rbrakk> \<Longrightarrow>
```
```   169    height (balL l a r) = (1 + max (height l) (height r))"
```
```   170 by (cases l, cases r) (simp_all add: balL_def)
```
```   171
```
```   172 lemma height_balR2:
```
```   173   "\<lbrakk> avl l;  avl r;  height r \<noteq> height l + 2 \<rbrakk> \<Longrightarrow>
```
```   174    height (balR l a r) = (1 + max (height l) (height r))"
```
```   175 by (cases l, cases r) (simp_all add: balR_def)
```
```   176
```
```   177 lemma avl_balL:
```
```   178   assumes "avl l" "avl r" and "height l = height r \<or> height l = height r + 1
```
```   179     \<or> height r = height l + 1 \<or> height l = height r + 2"
```
```   180   shows "avl(balL l a r)"
```
```   181 proof(cases l)
```
```   182   case Leaf
```
```   183   with assms show ?thesis by (simp add: node_def balL_def)
```
```   184 next
```
```   185   case (Node ln ll lr lh)
```
```   186   with assms show ?thesis
```
```   187   proof(cases "height l = height r + 2")
```
```   188     case True
```
```   189     from True Node assms show ?thesis
```
```   190       by (auto simp: balL_def intro!: avl_node split: tree.split) arith+
```
```   191   next
```
```   192     case False
```
```   193     with assms show ?thesis by (simp add: avl_node balL_def)
```
```   194   qed
```
```   195 qed
```
```   196
```
```   197 lemma avl_balR:
```
```   198   assumes "avl l" and "avl r" and "height l = height r \<or> height l = height r + 1
```
```   199     \<or> height r = height l + 1 \<or> height r = height l + 2"
```
```   200   shows "avl(balR l a r)"
```
```   201 proof(cases r)
```
```   202   case Leaf
```
```   203   with assms show ?thesis by (simp add: node_def balR_def)
```
```   204 next
```
```   205   case (Node rn rl rr rh)
```
```   206   with assms show ?thesis
```
```   207   proof(cases "height r = height l + 2")
```
```   208     case True
```
```   209       from True Node assms show ?thesis
```
```   210         by (auto simp: balR_def intro!: avl_node split: tree.split) arith+
```
```   211   next
```
```   212     case False
```
```   213     with assms show ?thesis by (simp add: balR_def avl_node)
```
```   214   qed
```
```   215 qed
```
```   216
```
```   217 (* It appears that these two properties need to be proved simultaneously: *)
```
```   218
```
```   219 text{* Insertion maintains the AVL property: *}
```
```   220
```
```   221 theorem avl_insert_aux:
```
```   222   assumes "avl t"
```
```   223   shows "avl(insert x t)"
```
```   224         "(height (insert x t) = height t \<or> height (insert x t) = height t + 1)"
```
```   225 using assms
```
```   226 proof (induction t)
```
```   227   case (Node h l a r)
```
```   228   case 1
```
```   229   with Node show ?case
```
```   230   proof(cases "x = a")
```
```   231     case True
```
```   232     with Node 1 show ?thesis by simp
```
```   233   next
```
```   234     case False
```
```   235     with Node 1 show ?thesis
```
```   236     proof(cases "x<a")
```
```   237       case True
```
```   238       with Node 1 show ?thesis by (auto simp add:avl_balL)
```
```   239     next
```
```   240       case False
```
```   241       with Node 1 `x\<noteq>a` show ?thesis by (auto simp add:avl_balR)
```
```   242     qed
```
```   243   qed
```
```   244   case 2
```
```   245   from 2 Node show ?case
```
```   246   proof(cases "x = a")
```
```   247     case True
```
```   248     with Node 1 show ?thesis by simp
```
```   249   next
```
```   250     case False
```
```   251     with Node 1 show ?thesis
```
```   252      proof(cases "x<a")
```
```   253       case True
```
```   254       with Node 2 show ?thesis
```
```   255       proof(cases "height (insert x l) = height r + 2")
```
```   256         case False with Node 2 `x < a` show ?thesis by (auto simp: height_balL2)
```
```   257       next
```
```   258         case True
```
```   259         hence "(height (balL (insert x l) a r) = height r + 2) \<or>
```
```   260           (height (balL (insert x l) a r) = height r + 3)" (is "?A \<or> ?B")
```
```   261           using Node 2 by (intro height_balL) simp_all
```
```   262         thus ?thesis
```
```   263         proof
```
```   264           assume ?A
```
```   265           with 2 `x < a` show ?thesis by (auto)
```
```   266         next
```
```   267           assume ?B
```
```   268           with True 1 Node(2) `x < a` show ?thesis by (simp) arith
```
```   269         qed
```
```   270       qed
```
```   271     next
```
```   272       case False
```
```   273       with Node 2 show ?thesis
```
```   274       proof(cases "height (insert x r) = height l + 2")
```
```   275         case False
```
```   276         with Node 2 `\<not>x < a` show ?thesis by (auto simp: height_balR2)
```
```   277       next
```
```   278         case True
```
```   279         hence "(height (balR l a (insert x r)) = height l + 2) \<or>
```
```   280           (height (balR l a (insert x r)) = height l + 3)"  (is "?A \<or> ?B")
```
```   281           using Node 2 by (intro height_balR) simp_all
```
```   282         thus ?thesis
```
```   283         proof
```
```   284           assume ?A
```
```   285           with 2 `\<not>x < a` show ?thesis by (auto)
```
```   286         next
```
```   287           assume ?B
```
```   288           with True 1 Node(4) `\<not>x < a` show ?thesis by (simp) arith
```
```   289         qed
```
```   290       qed
```
```   291     qed
```
```   292   qed
```
```   293 qed simp_all
```
```   294
```
```   295
```
```   296 subsubsection {* Deletion maintains AVL balance *}
```
```   297
```
```   298 lemma avl_delete_max:
```
```   299   assumes "avl x" and "x \<noteq> Leaf"
```
```   300   shows "avl (fst (delete_max x))" "height x = height(fst (delete_max x)) \<or>
```
```   301          height x = height(fst (delete_max x)) + 1"
```
```   302 using assms
```
```   303 proof (induct x rule: delete_max_induct)
```
```   304   case (Node h l a rh rl b rr)
```
```   305   case 1
```
```   306   with Node have "avl l" "avl (fst (delete_max (Node rh rl b rr)))" by auto
```
```   307   with 1 Node have "avl (balL l a (fst (delete_max (Node rh rl b rr))))"
```
```   308     by (intro avl_balL) fastforce+
```
```   309   thus ?case
```
```   310     by (auto simp: height_balL height_balL2
```
```   311       linorder_class.max.absorb1 linorder_class.max.absorb2
```
```   312       split:prod.split)
```
```   313 next
```
```   314   case (Node h l a rh rl b rr)
```
```   315   case 2
```
```   316   let ?r = "Node rh rl b rr"
```
```   317   let ?r' = "fst (delete_max ?r)"
```
```   318   from `avl x` Node 2 have "avl l" and "avl ?r" by simp_all
```
```   319   thus ?case using Node 2 height_balL[of l ?r' a] height_balL2[of l ?r' a]
```
```   320     apply (auto split:prod.splits simp del:avl.simps) by arith+
```
```   321 qed auto
```
```   322
```
```   323 lemma avl_delete_root:
```
```   324   assumes "avl t" and "t \<noteq> Leaf"
```
```   325   shows "avl(delete_root t)"
```
```   326 using assms
```
```   327 proof (cases t rule:delete_root_cases)
```
```   328   case (Node_Node h lh ll ln lr n rh rl rn rr)
```
```   329   let ?l = "Node lh ll ln lr"
```
```   330   let ?r = "Node rh rl rn rr"
```
```   331   let ?l' = "fst (delete_max ?l)"
```
```   332   from `avl t` and Node_Node have "avl ?r" by simp
```
```   333   from `avl t` and Node_Node have "avl ?l" by simp
```
```   334   hence "avl(?l')" "height ?l = height(?l') \<or>
```
```   335          height ?l = height(?l') + 1" by (rule avl_delete_max,simp)+
```
```   336   with `avl t` Node_Node have "height ?l' = height ?r \<or> height ?l' = height ?r + 1
```
```   337             \<or> height ?r = height ?l' + 1 \<or> height ?r = height ?l' + 2" by fastforce
```
```   338   with `avl ?l'` `avl ?r` have "avl(balR ?l' (snd(delete_max ?l)) ?r)"
```
```   339     by (rule avl_balR)
```
```   340   with Node_Node show ?thesis by (auto split:prod.splits)
```
```   341 qed simp_all
```
```   342
```
```   343 lemma height_delete_root:
```
```   344   assumes "avl t" and "t \<noteq> Leaf"
```
```   345   shows "height t = height(delete_root t) \<or> height t = height(delete_root t) + 1"
```
```   346 using assms
```
```   347 proof (cases t rule: delete_root_cases)
```
```   348   case (Node_Node h lh ll ln lr n rh rl rn rr)
```
```   349   let ?l = "Node lh ll ln lr"
```
```   350   let ?r = "Node rh rl rn rr"
```
```   351   let ?l' = "fst (delete_max ?l)"
```
```   352   let ?t' = "balR ?l' (snd(delete_max ?l)) ?r"
```
```   353   from `avl t` and Node_Node have "avl ?r" by simp
```
```   354   from `avl t` and Node_Node have "avl ?l" by simp
```
```   355   hence "avl(?l')"  by (rule avl_delete_max,simp)
```
```   356   have l'_height: "height ?l = height ?l' \<or> height ?l = height ?l' + 1" using `avl ?l` by (intro avl_delete_max) auto
```
```   357   have t_height: "height t = 1 + max (height ?l) (height ?r)" using `avl t` Node_Node by simp
```
```   358   have "height t = height ?t' \<or> height t = height ?t' + 1" using  `avl t` Node_Node
```
```   359   proof(cases "height ?r = height ?l' + 2")
```
```   360     case False
```
```   361     show ?thesis using l'_height t_height False by (subst  height_balR2[OF `avl ?l'` `avl ?r` False])+ arith
```
```   362   next
```
```   363     case True
```
```   364     show ?thesis
```
```   365     proof(cases rule: disjE[OF height_balR[OF True `avl ?l'` `avl ?r`, of "snd (delete_max ?l)"]])
```
```   366       case 1
```
```   367       thus ?thesis using l'_height t_height True by arith
```
```   368     next
```
```   369       case 2
```
```   370       thus ?thesis using l'_height t_height True by arith
```
```   371     qed
```
```   372   qed
```
```   373   thus ?thesis using Node_Node by (auto split:prod.splits)
```
```   374 qed simp_all
```
```   375
```
```   376 text{* Deletion maintains the AVL property: *}
```
```   377
```
```   378 theorem avl_delete_aux:
```
```   379   assumes "avl t"
```
```   380   shows "avl(delete x t)" and "height t = (height (delete x t)) \<or> height t = height (delete x t) + 1"
```
```   381 using assms
```
```   382 proof (induct t)
```
```   383   case (Node h l n r)
```
```   384   case 1
```
```   385   with Node show ?case
```
```   386   proof(cases "x = n")
```
```   387     case True
```
```   388     with Node 1 show ?thesis by (auto simp:avl_delete_root)
```
```   389   next
```
```   390     case False
```
```   391     with Node 1 show ?thesis
```
```   392     proof(cases "x<n")
```
```   393       case True
```
```   394       with Node 1 show ?thesis by (auto simp add:avl_balR)
```
```   395     next
```
```   396       case False
```
```   397       with Node 1 `x\<noteq>n` show ?thesis by (auto simp add:avl_balL)
```
```   398     qed
```
```   399   qed
```
```   400   case 2
```
```   401   with Node show ?case
```
```   402   proof(cases "x = n")
```
```   403     case True
```
```   404     with 1 have "height (Node h l n r) = height(delete_root (Node h l n r))
```
```   405       \<or> height (Node h l n r) = height(delete_root (Node h l n r)) + 1"
```
```   406       by (subst height_delete_root,simp_all)
```
```   407     with True show ?thesis by simp
```
```   408   next
```
```   409     case False
```
```   410     with Node 1 show ?thesis
```
```   411      proof(cases "x<n")
```
```   412       case True
```
```   413       show ?thesis
```
```   414       proof(cases "height r = height (delete x l) + 2")
```
```   415         case False with Node 1 `x < n` show ?thesis by(auto simp: balR_def)
```
```   416       next
```
```   417         case True
```
```   418         hence "(height (balR (delete x l) n r) = height (delete x l) + 2) \<or>
```
```   419           height (balR (delete x l) n r) = height (delete x l) + 3" (is "?A \<or> ?B")
```
```   420           using Node 2 by (intro height_balR) auto
```
```   421         thus ?thesis
```
```   422         proof
```
```   423           assume ?A
```
```   424           with `x < n` Node 2 show ?thesis by(auto simp: balR_def)
```
```   425         next
```
```   426           assume ?B
```
```   427           with `x < n` Node 2 show ?thesis by(auto simp: balR_def)
```
```   428         qed
```
```   429       qed
```
```   430     next
```
```   431       case False
```
```   432       show ?thesis
```
```   433       proof(cases "height l = height (delete x r) + 2")
```
```   434         case False with Node 1 `\<not>x < n` `x \<noteq> n` show ?thesis by(auto simp: balL_def)
```
```   435       next
```
```   436         case True
```
```   437         hence "(height (balL l n (delete x r)) = height (delete x r) + 2) \<or>
```
```   438           height (balL l n (delete x r)) = height (delete x r) + 3" (is "?A \<or> ?B")
```
```   439           using Node 2 by (intro height_balL) auto
```
```   440         thus ?thesis
```
```   441         proof
```
```   442           assume ?A
```
```   443           with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: balL_def)
```
```   444         next
```
```   445           assume ?B
```
```   446           with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: balL_def)
```
```   447         qed
```
```   448       qed
```
```   449     qed
```
```   450   qed
```
```   451 qed simp_all
```
```   452
```
```   453 end
```