src/HOL/Data_Structures/AVL_Set.thy
 author nipkow Fri Nov 13 12:06:50 2015 +0100 (2015-11-13) changeset 61647 5121b9a57cce parent 61588 1d2907d0ed73 child 61678 b594e9277be3 permissions -rw-r--r--
tuned
```     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 del_max :: "'a avl_tree \<Rightarrow> 'a avl_tree * 'a" where
```
```    55 "del_max (Node _ l a r) = (if r = Leaf then (l,a)
```
```    56   else let (r',a') = del_max r in (balL l a r', a'))"
```
```    57
```
```    58 lemmas del_max_induct = del_max.induct[case_names Node Leaf]
```
```    59
```
```    60 fun del_root :: "'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    61 "del_root (Node h Leaf a r) = r" |
```
```    62 "del_root (Node h l a Leaf) = l" |
```
```    63 "del_root (Node h l a r) = (let (l', a') = del_max l in balR l' a' r)"
```
```    64
```
```    65 lemmas del_root_cases = del_root.cases[case_names Leaf_t Node_Leaf Node_Node]
```
```    66
```
```    67 fun delete :: "'a::cmp \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    68 "delete _ Leaf = Leaf" |
```
```    69 "delete x (Node h l a r) = (case cmp x a of
```
```    70    EQ \<Rightarrow> del_root (Node h l a r) |
```
```    71    LT \<Rightarrow> balR (delete x l) a r |
```
```    72    GT \<Rightarrow> balL l a (delete x r))"
```
```    73
```
```    74
```
```    75 subsection {* Functional Correctness Proofs *}
```
```    76
```
```    77 text{* Very different from the AFP/AVL proofs *}
```
```    78
```
```    79
```
```    80 subsubsection "Proofs for insert"
```
```    81
```
```    82 lemma inorder_balL:
```
```    83   "inorder (balL l a r) = inorder l @ a # inorder r"
```
```    84 by (auto simp: node_def balL_def split:tree.splits)
```
```    85
```
```    86 lemma inorder_balR:
```
```    87   "inorder (balR l a r) = inorder l @ a # inorder r"
```
```    88 by (auto simp: node_def balR_def split:tree.splits)
```
```    89
```
```    90 theorem inorder_insert:
```
```    91   "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
```
```    92 by (induct t)
```
```    93    (auto simp: ins_list_simps inorder_balL inorder_balR)
```
```    94
```
```    95
```
```    96 subsubsection "Proofs for delete"
```
```    97
```
```    98 lemma inorder_del_maxD:
```
```    99   "\<lbrakk> del_max t = (t',a); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
```
```   100    inorder t' @ [a] = inorder t"
```
```   101 by(induction t arbitrary: t' rule: del_max.induct)
```
```   102   (auto simp: inorder_balL split: if_splits prod.splits tree.split)
```
```   103
```
```   104 lemma inorder_del_root:
```
```   105   "inorder (del_root (Node h l a r)) = inorder l @ inorder r"
```
```   106 by(induction "Node h l a r" arbitrary: l a r h rule: del_root.induct)
```
```   107   (auto simp: inorder_balL inorder_balR inorder_del_maxD split: if_splits prod.splits)
```
```   108
```
```   109 theorem inorder_delete:
```
```   110   "sorted(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
```
```   111 by(induction t)
```
```   112   (auto simp: del_list_simps inorder_balL inorder_balR
```
```   113     inorder_del_root inorder_del_maxD split: prod.splits)
```
```   114
```
```   115
```
```   116 subsubsection "Overall functional correctness"
```
```   117
```
```   118 interpretation Set_by_Ordered
```
```   119 where empty = Leaf and isin = isin and insert = insert and delete = delete
```
```   120 and inorder = inorder and inv = "\<lambda>_. True"
```
```   121 proof (standard, goal_cases)
```
```   122   case 1 show ?case by simp
```
```   123 next
```
```   124   case 2 thus ?case by(simp add: isin_set)
```
```   125 next
```
```   126   case 3 thus ?case by(simp add: inorder_insert)
```
```   127 next
```
```   128   case 4 thus ?case by(simp add: inorder_delete)
```
```   129 qed (rule TrueI)+
```
```   130
```
```   131
```
```   132 subsection {* AVL invariants *}
```
```   133
```
```   134 text{* Essentially the AFP/AVL proofs *}
```
```   135
```
```   136
```
```   137 subsubsection {* Insertion maintains AVL balance *}
```
```   138
```
```   139 declare Let_def [simp]
```
```   140
```
```   141 lemma [simp]: "avl t \<Longrightarrow> ht t = height t"
```
```   142 by (induct t) simp_all
```
```   143
```
```   144 lemma height_balL:
```
```   145   "\<lbrakk> height l = height r + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
```
```   146    height (balL l a r) = height r + 2 \<or>
```
```   147    height (balL l a r) = height r + 3"
```
```   148 by (cases l) (auto simp:node_def balL_def split:tree.split)
```
```   149
```
```   150 lemma height_balR:
```
```   151   "\<lbrakk> height r = height l + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
```
```   152    height (balR l a r) = height l + 2 \<or>
```
```   153    height (balR l a r) = height l + 3"
```
```   154 by (cases r) (auto simp add:node_def balR_def split:tree.split)
```
```   155
```
```   156 lemma [simp]: "height(node l a r) = max (height l) (height r) + 1"
```
```   157 by (simp add: node_def)
```
```   158
```
```   159 lemma avl_node:
```
```   160   "\<lbrakk> avl l; avl r;
```
```   161      height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1
```
```   162    \<rbrakk> \<Longrightarrow> avl(node l a r)"
```
```   163 by (auto simp add:max_def node_def)
```
```   164
```
```   165 lemma height_balL2:
```
```   166   "\<lbrakk> avl l; avl r; height l \<noteq> height r + 2 \<rbrakk> \<Longrightarrow>
```
```   167    height (balL l a r) = (1 + max (height l) (height r))"
```
```   168 by (cases l, cases r) (simp_all add: balL_def)
```
```   169
```
```   170 lemma height_balR2:
```
```   171   "\<lbrakk> avl l;  avl r;  height r \<noteq> height l + 2 \<rbrakk> \<Longrightarrow>
```
```   172    height (balR l a r) = (1 + max (height l) (height r))"
```
```   173 by (cases l, cases r) (simp_all add: balR_def)
```
```   174
```
```   175 lemma avl_balL:
```
```   176   assumes "avl l" "avl r" and "height l = height r \<or> height l = height r + 1
```
```   177     \<or> height r = height l + 1 \<or> height l = height r + 2"
```
```   178   shows "avl(balL l a r)"
```
```   179 proof(cases l)
```
```   180   case Leaf
```
```   181   with assms show ?thesis by (simp add: node_def balL_def)
```
```   182 next
```
```   183   case (Node ln ll lr lh)
```
```   184   with assms show ?thesis
```
```   185   proof(cases "height l = height r + 2")
```
```   186     case True
```
```   187     from True Node assms show ?thesis
```
```   188       by (auto simp: balL_def intro!: avl_node split: tree.split) arith+
```
```   189   next
```
```   190     case False
```
```   191     with assms show ?thesis by (simp add: avl_node balL_def)
```
```   192   qed
```
```   193 qed
```
```   194
```
```   195 lemma avl_balR:
```
```   196   assumes "avl l" and "avl r" and "height l = height r \<or> height l = height r + 1
```
```   197     \<or> height r = height l + 1 \<or> height r = height l + 2"
```
```   198   shows "avl(balR l a r)"
```
```   199 proof(cases r)
```
```   200   case Leaf
```
```   201   with assms show ?thesis by (simp add: node_def balR_def)
```
```   202 next
```
```   203   case (Node rn rl rr rh)
```
```   204   with assms show ?thesis
```
```   205   proof(cases "height r = height l + 2")
```
```   206     case True
```
```   207       from True Node assms show ?thesis
```
```   208         by (auto simp: balR_def intro!: avl_node split: tree.split) arith+
```
```   209   next
```
```   210     case False
```
```   211     with assms show ?thesis by (simp add: balR_def avl_node)
```
```   212   qed
```
```   213 qed
```
```   214
```
```   215 (* It appears that these two properties need to be proved simultaneously: *)
```
```   216
```
```   217 text{* Insertion maintains the AVL property: *}
```
```   218
```
```   219 theorem avl_insert_aux:
```
```   220   assumes "avl t"
```
```   221   shows "avl(insert x t)"
```
```   222         "(height (insert x t) = height t \<or> height (insert x t) = height t + 1)"
```
```   223 using assms
```
```   224 proof (induction t)
```
```   225   case (Node h l a r)
```
```   226   case 1
```
```   227   with Node show ?case
```
```   228   proof(cases "x = a")
```
```   229     case True
```
```   230     with Node 1 show ?thesis by simp
```
```   231   next
```
```   232     case False
```
```   233     with Node 1 show ?thesis
```
```   234     proof(cases "x<a")
```
```   235       case True
```
```   236       with Node 1 show ?thesis by (auto simp add:avl_balL)
```
```   237     next
```
```   238       case False
```
```   239       with Node 1 `x\<noteq>a` show ?thesis by (auto simp add:avl_balR)
```
```   240     qed
```
```   241   qed
```
```   242   case 2
```
```   243   from 2 Node show ?case
```
```   244   proof(cases "x = a")
```
```   245     case True
```
```   246     with Node 1 show ?thesis by simp
```
```   247   next
```
```   248     case False
```
```   249     with Node 1 show ?thesis
```
```   250      proof(cases "x<a")
```
```   251       case True
```
```   252       with Node 2 show ?thesis
```
```   253       proof(cases "height (insert x l) = height r + 2")
```
```   254         case False with Node 2 `x < a` show ?thesis by (auto simp: height_balL2)
```
```   255       next
```
```   256         case True
```
```   257         hence "(height (balL (insert x l) a r) = height r + 2) \<or>
```
```   258           (height (balL (insert x l) a r) = height r + 3)" (is "?A \<or> ?B")
```
```   259           using Node 2 by (intro height_balL) simp_all
```
```   260         thus ?thesis
```
```   261         proof
```
```   262           assume ?A
```
```   263           with 2 `x < a` show ?thesis by (auto)
```
```   264         next
```
```   265           assume ?B
```
```   266           with True 1 Node(2) `x < a` show ?thesis by (simp) arith
```
```   267         qed
```
```   268       qed
```
```   269     next
```
```   270       case False
```
```   271       with Node 2 show ?thesis
```
```   272       proof(cases "height (insert x r) = height l + 2")
```
```   273         case False
```
```   274         with Node 2 `\<not>x < a` show ?thesis by (auto simp: height_balR2)
```
```   275       next
```
```   276         case True
```
```   277         hence "(height (balR l a (insert x r)) = height l + 2) \<or>
```
```   278           (height (balR l a (insert x r)) = height l + 3)"  (is "?A \<or> ?B")
```
```   279           using Node 2 by (intro height_balR) simp_all
```
```   280         thus ?thesis
```
```   281         proof
```
```   282           assume ?A
```
```   283           with 2 `\<not>x < a` show ?thesis by (auto)
```
```   284         next
```
```   285           assume ?B
```
```   286           with True 1 Node(4) `\<not>x < a` show ?thesis by (simp) arith
```
```   287         qed
```
```   288       qed
```
```   289     qed
```
```   290   qed
```
```   291 qed simp_all
```
```   292
```
```   293
```
```   294 subsubsection {* Deletion maintains AVL balance *}
```
```   295
```
```   296 lemma avl_del_max:
```
```   297   assumes "avl x" and "x \<noteq> Leaf"
```
```   298   shows "avl (fst (del_max x))" "height x = height(fst (del_max x)) \<or>
```
```   299          height x = height(fst (del_max x)) + 1"
```
```   300 using assms
```
```   301 proof (induct x rule: del_max_induct)
```
```   302   case (Node h l a r)
```
```   303   case 1
```
```   304   thus ?case using Node
```
```   305     by (auto simp: height_balL height_balL2 avl_balL
```
```   306       linorder_class.max.absorb1 linorder_class.max.absorb2
```
```   307       split:prod.split)
```
```   308 next
```
```   309   case (Node h l a r)
```
```   310   case 2
```
```   311   let ?r' = "fst (del_max r)"
```
```   312   from `avl x` Node 2 have "avl l" and "avl r" by simp_all
```
```   313   thus ?case using Node 2 height_balL[of l ?r' a] height_balL2[of l ?r' a]
```
```   314     apply (auto split:prod.splits simp del:avl.simps) by arith+
```
```   315 qed auto
```
```   316
```
```   317 lemma avl_del_root:
```
```   318   assumes "avl t" and "t \<noteq> Leaf"
```
```   319   shows "avl(del_root t)"
```
```   320 using assms
```
```   321 proof (cases t rule:del_root_cases)
```
```   322   case (Node_Node h lh ll ln lr n rh rl rn rr)
```
```   323   let ?l = "Node lh ll ln lr"
```
```   324   let ?r = "Node rh rl rn rr"
```
```   325   let ?l' = "fst (del_max ?l)"
```
```   326   from `avl t` and Node_Node have "avl ?r" by simp
```
```   327   from `avl t` and Node_Node have "avl ?l" by simp
```
```   328   hence "avl(?l')" "height ?l = height(?l') \<or>
```
```   329          height ?l = height(?l') + 1" by (rule avl_del_max,simp)+
```
```   330   with `avl t` Node_Node have "height ?l' = height ?r \<or> height ?l' = height ?r + 1
```
```   331             \<or> height ?r = height ?l' + 1 \<or> height ?r = height ?l' + 2" by fastforce
```
```   332   with `avl ?l'` `avl ?r` have "avl(balR ?l' (snd(del_max ?l)) ?r)"
```
```   333     by (rule avl_balR)
```
```   334   with Node_Node show ?thesis by (auto split:prod.splits)
```
```   335 qed simp_all
```
```   336
```
```   337 lemma height_del_root:
```
```   338   assumes "avl t" and "t \<noteq> Leaf"
```
```   339   shows "height t = height(del_root t) \<or> height t = height(del_root t) + 1"
```
```   340 using assms
```
```   341 proof (cases t rule: del_root_cases)
```
```   342   case (Node_Node h lh ll ln lr n rh rl rn rr)
```
```   343   let ?l = "Node lh ll ln lr"
```
```   344   let ?r = "Node rh rl rn rr"
```
```   345   let ?l' = "fst (del_max ?l)"
```
```   346   let ?t' = "balR ?l' (snd(del_max ?l)) ?r"
```
```   347   from `avl t` and Node_Node have "avl ?r" by simp
```
```   348   from `avl t` and Node_Node have "avl ?l" by simp
```
```   349   hence "avl(?l')"  by (rule avl_del_max,simp)
```
```   350   have l'_height: "height ?l = height ?l' \<or> height ?l = height ?l' + 1" using `avl ?l` by (intro avl_del_max) auto
```
```   351   have t_height: "height t = 1 + max (height ?l) (height ?r)" using `avl t` Node_Node by simp
```
```   352   have "height t = height ?t' \<or> height t = height ?t' + 1" using  `avl t` Node_Node
```
```   353   proof(cases "height ?r = height ?l' + 2")
```
```   354     case False
```
```   355     show ?thesis using l'_height t_height False by (subst  height_balR2[OF `avl ?l'` `avl ?r` False])+ arith
```
```   356   next
```
```   357     case True
```
```   358     show ?thesis
```
```   359     proof(cases rule: disjE[OF height_balR[OF True `avl ?l'` `avl ?r`, of "snd (del_max ?l)"]])
```
```   360       case 1
```
```   361       thus ?thesis using l'_height t_height True by arith
```
```   362     next
```
```   363       case 2
```
```   364       thus ?thesis using l'_height t_height True by arith
```
```   365     qed
```
```   366   qed
```
```   367   thus ?thesis using Node_Node by (auto split:prod.splits)
```
```   368 qed simp_all
```
```   369
```
```   370 text{* Deletion maintains the AVL property: *}
```
```   371
```
```   372 theorem avl_delete_aux:
```
```   373   assumes "avl t"
```
```   374   shows "avl(delete x t)" and "height t = (height (delete x t)) \<or> height t = height (delete x t) + 1"
```
```   375 using assms
```
```   376 proof (induct t)
```
```   377   case (Node h l n r)
```
```   378   case 1
```
```   379   with Node show ?case
```
```   380   proof(cases "x = n")
```
```   381     case True
```
```   382     with Node 1 show ?thesis by (auto simp:avl_del_root)
```
```   383   next
```
```   384     case False
```
```   385     with Node 1 show ?thesis
```
```   386     proof(cases "x<n")
```
```   387       case True
```
```   388       with Node 1 show ?thesis by (auto simp add:avl_balR)
```
```   389     next
```
```   390       case False
```
```   391       with Node 1 `x\<noteq>n` show ?thesis by (auto simp add:avl_balL)
```
```   392     qed
```
```   393   qed
```
```   394   case 2
```
```   395   with Node show ?case
```
```   396   proof(cases "x = n")
```
```   397     case True
```
```   398     with 1 have "height (Node h l n r) = height(del_root (Node h l n r))
```
```   399       \<or> height (Node h l n r) = height(del_root (Node h l n r)) + 1"
```
```   400       by (subst height_del_root,simp_all)
```
```   401     with True show ?thesis by simp
```
```   402   next
```
```   403     case False
```
```   404     with Node 1 show ?thesis
```
```   405      proof(cases "x<n")
```
```   406       case True
```
```   407       show ?thesis
```
```   408       proof(cases "height r = height (delete x l) + 2")
```
```   409         case False with Node 1 `x < n` show ?thesis by(auto simp: balR_def)
```
```   410       next
```
```   411         case True
```
```   412         hence "(height (balR (delete x l) n r) = height (delete x l) + 2) \<or>
```
```   413           height (balR (delete x l) n r) = height (delete x l) + 3" (is "?A \<or> ?B")
```
```   414           using Node 2 by (intro height_balR) auto
```
```   415         thus ?thesis
```
```   416         proof
```
```   417           assume ?A
```
```   418           with `x < n` Node 2 show ?thesis by(auto simp: balR_def)
```
```   419         next
```
```   420           assume ?B
```
```   421           with `x < n` Node 2 show ?thesis by(auto simp: balR_def)
```
```   422         qed
```
```   423       qed
```
```   424     next
```
```   425       case False
```
```   426       show ?thesis
```
```   427       proof(cases "height l = height (delete x r) + 2")
```
```   428         case False with Node 1 `\<not>x < n` `x \<noteq> n` show ?thesis by(auto simp: balL_def)
```
```   429       next
```
```   430         case True
```
```   431         hence "(height (balL l n (delete x r)) = height (delete x r) + 2) \<or>
```
```   432           height (balL l n (delete x r)) = height (delete x r) + 3" (is "?A \<or> ?B")
```
```   433           using Node 2 by (intro height_balL) auto
```
```   434         thus ?thesis
```
```   435         proof
```
```   436           assume ?A
```
```   437           with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: balL_def)
```
```   438         next
```
```   439           assume ?B
```
```   440           with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: balL_def)
```
```   441         qed
```
```   442       qed
```
```   443     qed
```
```   444   qed
```
```   445 qed simp_all
```
```   446
```
```   447 end
```