src/HOL/Data_Structures/AVL_Set.thy
 author nipkow Tue Oct 13 17:06:37 2015 +0200 (2015-10-13) changeset 61428 5e1938107371 parent 61232 c46faf9762f7 child 61581 00d9682e8dd7 permissions -rw-r--r--
added invar empty
```     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 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 node_bal_l :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    30 "node_bal_l 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 node_bal_r :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    39 "node_bal_r 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::order \<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) =
```
```    50    (if x=a then Node h l a r
```
```    51     else if x<a
```
```    52       then node_bal_l (insert x l) a r
```
```    53       else node_bal_r l a (insert x r))"
```
```    54
```
```    55 fun delete_max :: "'a avl_tree \<Rightarrow> 'a avl_tree * 'a" where
```
```    56 "delete_max (Node _ l a Leaf) = (l,a)" |
```
```    57 "delete_max (Node _ l a r) = (
```
```    58   let (r',a') = delete_max r in
```
```    59   (node_bal_l l a r', a'))"
```
```    60
```
```    61 lemmas delete_max_induct = delete_max.induct[case_names Leaf Node]
```
```    62
```
```    63 fun delete_root :: "'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    64 "delete_root (Node h Leaf a r) = r" |
```
```    65 "delete_root (Node h l a Leaf) = l" |
```
```    66 "delete_root (Node h l a r) =
```
```    67   (let (l', a') = delete_max l in node_bal_r l' a' r)"
```
```    68
```
```    69 lemmas delete_root_cases = delete_root.cases[case_names Leaf_t Node_Leaf Node_Node]
```
```    70
```
```    71 fun delete :: "'a::order \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
```
```    72 "delete _ Leaf = Leaf" |
```
```    73 "delete x (Node h l a r) = (
```
```    74    if x = a then delete_root (Node h l a r)
```
```    75    else if x < a then node_bal_r (delete x l) a r
```
```    76    else node_bal_l l a (delete x r))"
```
```    77
```
```    78
```
```    79 subsection {* Functional Correctness Proofs *}
```
```    80
```
```    81 text{* Very different from the AFP/AVL proofs *}
```
```    82
```
```    83
```
```    84 subsubsection "Proofs for insert"
```
```    85
```
```    86 lemma inorder_node_bal_l:
```
```    87   "inorder (node_bal_l l a r) = inorder l @ a # inorder r"
```
```    88 by (auto simp: node_def node_bal_l_def split:tree.splits)
```
```    89
```
```    90 lemma inorder_node_bal_r:
```
```    91   "inorder (node_bal_r l a r) = inorder l @ a # inorder r"
```
```    92 by (auto simp: node_def node_bal_r_def split:tree.splits)
```
```    93
```
```    94 theorem inorder_insert:
```
```    95   "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
```
```    96 by (induct t)
```
```    97    (auto simp: ins_list_simps inorder_node_bal_l inorder_node_bal_r)
```
```    98
```
```    99
```
```   100 subsubsection "Proofs for delete"
```
```   101
```
```   102 lemma inorder_delete_maxD:
```
```   103   "\<lbrakk> delete_max t = (t',a); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
```
```   104    inorder t' @ [a] = inorder t"
```
```   105 by(induction t arbitrary: t' rule: delete_max.induct)
```
```   106   (auto simp: inorder_node_bal_l split: prod.splits tree.split)
```
```   107
```
```   108 lemma inorder_delete_root:
```
```   109   "inorder (delete_root (Node h l a r)) = inorder l @ inorder r"
```
```   110 by(induction "Node h l a r" arbitrary: l a r h rule: delete_root.induct)
```
```   111   (auto simp: inorder_node_bal_r inorder_delete_maxD split: prod.splits)
```
```   112
```
```   113 theorem inorder_delete:
```
```   114   "sorted(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
```
```   115 by(induction t)
```
```   116   (auto simp: del_list_simps inorder_node_bal_l inorder_node_bal_r
```
```   117     inorder_delete_root inorder_delete_maxD split: prod.splits)
```
```   118
```
```   119
```
```   120 subsubsection "Overall functional correctness"
```
```   121
```
```   122 interpretation Set_by_Ordered
```
```   123 where empty = Leaf and isin = isin and insert = insert and delete = delete
```
```   124 and inorder = inorder and wf = "\<lambda>_. True"
```
```   125 proof (standard, goal_cases)
```
```   126   case 1 show ?case by simp
```
```   127 next
```
```   128   case 2 thus ?case by(simp add: isin_set)
```
```   129 next
```
```   130   case 3 thus ?case by(simp add: inorder_insert)
```
```   131 next
```
```   132   case 4 thus ?case by(simp add: inorder_delete)
```
```   133 qed (rule TrueI)+
```
```   134
```
```   135
```
```   136 subsection {* AVL invariants *}
```
```   137
```
```   138 text{* Essentially the AFP/AVL proofs *}
```
```   139
```
```   140
```
```   141 subsubsection {* Insertion maintains AVL balance *}
```
```   142
```
```   143 declare Let_def [simp]
```
```   144
```
```   145 lemma [simp]: "avl t \<Longrightarrow> ht t = height t"
```
```   146 by (induct t) simp_all
```
```   147
```
```   148 lemma height_node_bal_l:
```
```   149   "\<lbrakk> height l = height r + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
```
```   150    height (node_bal_l l a r) = height r + 2 \<or>
```
```   151    height (node_bal_l l a r) = height r + 3"
```
```   152 by (cases l) (auto simp:node_def node_bal_l_def split:tree.split)
```
```   153
```
```   154 lemma height_node_bal_r:
```
```   155   "\<lbrakk> height r = height l + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
```
```   156    height (node_bal_r l a r) = height l + 2 \<or>
```
```   157    height (node_bal_r l a r) = height l + 3"
```
```   158 by (cases r) (auto simp add:node_def node_bal_r_def split:tree.split)
```
```   159
```
```   160 lemma [simp]: "height(node l a r) = max (height l) (height r) + 1"
```
```   161 by (simp add: node_def)
```
```   162
```
```   163 lemma avl_node:
```
```   164   "\<lbrakk> avl l; avl r;
```
```   165      height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1
```
```   166    \<rbrakk> \<Longrightarrow> avl(node l a r)"
```
```   167 by (auto simp add:max_def node_def)
```
```   168
```
```   169 lemma height_node_bal_l2:
```
```   170   "\<lbrakk> avl l; avl r; height l \<noteq> height r + 2 \<rbrakk> \<Longrightarrow>
```
```   171    height (node_bal_l l a r) = (1 + max (height l) (height r))"
```
```   172 by (cases l, cases r) (simp_all add: node_bal_l_def)
```
```   173
```
```   174 lemma height_node_bal_r2:
```
```   175   "\<lbrakk> avl l;  avl r;  height r \<noteq> height l + 2 \<rbrakk> \<Longrightarrow>
```
```   176    height (node_bal_r l a r) = (1 + max (height l) (height r))"
```
```   177 by (cases l, cases r) (simp_all add: node_bal_r_def)
```
```   178
```
```   179 lemma avl_node_bal_l:
```
```   180   assumes "avl l" "avl r" and "height l = height r \<or> height l = height r + 1
```
```   181     \<or> height r = height l + 1 \<or> height l = height r + 2"
```
```   182   shows "avl(node_bal_l l a r)"
```
```   183 proof(cases l)
```
```   184   case Leaf
```
```   185   with assms show ?thesis by (simp add: node_def node_bal_l_def)
```
```   186 next
```
```   187   case (Node ln ll lr lh)
```
```   188   with assms show ?thesis
```
```   189   proof(cases "height l = height r + 2")
```
```   190     case True
```
```   191     from True Node assms show ?thesis
```
```   192       by (auto simp: node_bal_l_def intro!: avl_node split: tree.split) arith+
```
```   193   next
```
```   194     case False
```
```   195     with assms show ?thesis by (simp add: avl_node node_bal_l_def)
```
```   196   qed
```
```   197 qed
```
```   198
```
```   199 lemma avl_node_bal_r:
```
```   200   assumes "avl l" and "avl r" and "height l = height r \<or> height l = height r + 1
```
```   201     \<or> height r = height l + 1 \<or> height r = height l + 2"
```
```   202   shows "avl(node_bal_r l a r)"
```
```   203 proof(cases r)
```
```   204   case Leaf
```
```   205   with assms show ?thesis by (simp add: node_def node_bal_r_def)
```
```   206 next
```
```   207   case (Node rn rl rr rh)
```
```   208   with assms show ?thesis
```
```   209   proof(cases "height r = height l + 2")
```
```   210     case True
```
```   211       from True Node assms show ?thesis
```
```   212         by (auto simp: node_bal_r_def intro!: avl_node split: tree.split) arith+
```
```   213   next
```
```   214     case False
```
```   215     with assms show ?thesis by (simp add: node_bal_r_def avl_node)
```
```   216   qed
```
```   217 qed
```
```   218
```
```   219 (* It appears that these two properties need to be proved simultaneously: *)
```
```   220
```
```   221 text{* Insertion maintains the AVL property: *}
```
```   222
```
```   223 theorem avl_insert_aux:
```
```   224   assumes "avl t"
```
```   225   shows "avl(insert x t)"
```
```   226         "(height (insert x t) = height t \<or> height (insert x t) = height t + 1)"
```
```   227 using assms
```
```   228 proof (induction t)
```
```   229   case (Node h l a r)
```
```   230   case 1
```
```   231   with Node show ?case
```
```   232   proof(cases "x = a")
```
```   233     case True
```
```   234     with Node 1 show ?thesis by simp
```
```   235   next
```
```   236     case False
```
```   237     with Node 1 show ?thesis
```
```   238     proof(cases "x<a")
```
```   239       case True
```
```   240       with Node 1 show ?thesis by (auto simp add:avl_node_bal_l)
```
```   241     next
```
```   242       case False
```
```   243       with Node 1 `x\<noteq>a` show ?thesis by (auto simp add:avl_node_bal_r)
```
```   244     qed
```
```   245   qed
```
```   246   case 2
```
```   247   from 2 Node show ?case
```
```   248   proof(cases "x = a")
```
```   249     case True
```
```   250     with Node 1 show ?thesis by simp
```
```   251   next
```
```   252     case False
```
```   253     with Node 1 show ?thesis
```
```   254      proof(cases "x<a")
```
```   255       case True
```
```   256       with Node 2 show ?thesis
```
```   257       proof(cases "height (insert x l) = height r + 2")
```
```   258         case False with Node 2 `x < a` show ?thesis by (auto simp: height_node_bal_l2)
```
```   259       next
```
```   260         case True
```
```   261         hence "(height (node_bal_l (insert x l) a r) = height r + 2) \<or>
```
```   262           (height (node_bal_l (insert x l) a r) = height r + 3)" (is "?A \<or> ?B")
```
```   263           using Node 2 by (intro height_node_bal_l) simp_all
```
```   264         thus ?thesis
```
```   265         proof
```
```   266           assume ?A
```
```   267           with 2 `x < a` show ?thesis by (auto)
```
```   268         next
```
```   269           assume ?B
```
```   270           with True 1 Node(2) `x < a` show ?thesis by (simp) arith
```
```   271         qed
```
```   272       qed
```
```   273     next
```
```   274       case False
```
```   275       with Node 2 show ?thesis
```
```   276       proof(cases "height (insert x r) = height l + 2")
```
```   277         case False
```
```   278         with Node 2 `\<not>x < a` show ?thesis by (auto simp: height_node_bal_r2)
```
```   279       next
```
```   280         case True
```
```   281         hence "(height (node_bal_r l a (insert x r)) = height l + 2) \<or>
```
```   282           (height (node_bal_r l a (insert x r)) = height l + 3)"  (is "?A \<or> ?B")
```
```   283           using Node 2 by (intro height_node_bal_r) simp_all
```
```   284         thus ?thesis
```
```   285         proof
```
```   286           assume ?A
```
```   287           with 2 `\<not>x < a` show ?thesis by (auto)
```
```   288         next
```
```   289           assume ?B
```
```   290           with True 1 Node(4) `\<not>x < a` show ?thesis by (simp) arith
```
```   291         qed
```
```   292       qed
```
```   293     qed
```
```   294   qed
```
```   295 qed simp_all
```
```   296
```
```   297
```
```   298 subsubsection {* Deletion maintains AVL balance *}
```
```   299
```
```   300 lemma avl_delete_max:
```
```   301   assumes "avl x" and "x \<noteq> Leaf"
```
```   302   shows "avl (fst (delete_max x))" "height x = height(fst (delete_max x)) \<or>
```
```   303          height x = height(fst (delete_max x)) + 1"
```
```   304 using assms
```
```   305 proof (induct x rule: delete_max_induct)
```
```   306   case (Node h l a rh rl b rr)
```
```   307   case 1
```
```   308   with Node have "avl l" "avl (fst (delete_max (Node rh rl b rr)))" by auto
```
```   309   with 1 Node have "avl (node_bal_l l a (fst (delete_max (Node rh rl b rr))))"
```
```   310     by (intro avl_node_bal_l) fastforce+
```
```   311   thus ?case
```
```   312     by (auto simp: height_node_bal_l height_node_bal_l2
```
```   313       linorder_class.max.absorb1 linorder_class.max.absorb2
```
```   314       split:prod.split)
```
```   315 next
```
```   316   case (Node h l a rh rl b rr)
```
```   317   case 2
```
```   318   let ?r = "Node rh rl b rr"
```
```   319   let ?r' = "fst (delete_max ?r)"
```
```   320   from `avl x` Node 2 have "avl l" and "avl ?r" by simp_all
```
```   321   thus ?case using Node 2 height_node_bal_l[of l ?r' a] height_node_bal_l2[of l ?r' a]
```
```   322     apply (auto split:prod.splits simp del:avl.simps) by arith+
```
```   323 qed auto
```
```   324
```
```   325 lemma avl_delete_root:
```
```   326   assumes "avl t" and "t \<noteq> Leaf"
```
```   327   shows "avl(delete_root t)"
```
```   328 using assms
```
```   329 proof (cases t rule:delete_root_cases)
```
```   330   case (Node_Node h lh ll ln lr n rh rl rn rr)
```
```   331   let ?l = "Node lh ll ln lr"
```
```   332   let ?r = "Node rh rl rn rr"
```
```   333   let ?l' = "fst (delete_max ?l)"
```
```   334   from `avl t` and Node_Node have "avl ?r" by simp
```
```   335   from `avl t` and Node_Node have "avl ?l" by simp
```
```   336   hence "avl(?l')" "height ?l = height(?l') \<or>
```
```   337          height ?l = height(?l') + 1" by (rule avl_delete_max,simp)+
```
```   338   with `avl t` Node_Node have "height ?l' = height ?r \<or> height ?l' = height ?r + 1
```
```   339             \<or> height ?r = height ?l' + 1 \<or> height ?r = height ?l' + 2" by fastforce
```
```   340   with `avl ?l'` `avl ?r` have "avl(node_bal_r ?l' (snd(delete_max ?l)) ?r)"
```
```   341     by (rule avl_node_bal_r)
```
```   342   with Node_Node show ?thesis by (auto split:prod.splits)
```
```   343 qed simp_all
```
```   344
```
```   345 lemma height_delete_root:
```
```   346   assumes "avl t" and "t \<noteq> Leaf"
```
```   347   shows "height t = height(delete_root t) \<or> height t = height(delete_root t) + 1"
```
```   348 using assms
```
```   349 proof (cases t rule: delete_root_cases)
```
```   350   case (Node_Node h lh ll ln lr n rh rl rn rr)
```
```   351   let ?l = "Node lh ll ln lr"
```
```   352   let ?r = "Node rh rl rn rr"
```
```   353   let ?l' = "fst (delete_max ?l)"
```
```   354   let ?t' = "node_bal_r ?l' (snd(delete_max ?l)) ?r"
```
```   355   from `avl t` and Node_Node have "avl ?r" by simp
```
```   356   from `avl t` and Node_Node have "avl ?l" by simp
```
```   357   hence "avl(?l')"  by (rule avl_delete_max,simp)
```
```   358   have l'_height: "height ?l = height ?l' \<or> height ?l = height ?l' + 1" using `avl ?l` by (intro avl_delete_max) auto
```
```   359   have t_height: "height t = 1 + max (height ?l) (height ?r)" using `avl t` Node_Node by simp
```
```   360   have "height t = height ?t' \<or> height t = height ?t' + 1" using  `avl t` Node_Node
```
```   361   proof(cases "height ?r = height ?l' + 2")
```
```   362     case False
```
```   363     show ?thesis using l'_height t_height False by (subst  height_node_bal_r2[OF `avl ?l'` `avl ?r` False])+ arith
```
```   364   next
```
```   365     case True
```
```   366     show ?thesis
```
```   367     proof(cases rule: disjE[OF height_node_bal_r[OF True `avl ?l'` `avl ?r`, of "snd (delete_max ?l)"]])
```
```   368       case 1
```
```   369       thus ?thesis using l'_height t_height True by arith
```
```   370     next
```
```   371       case 2
```
```   372       thus ?thesis using l'_height t_height True by arith
```
```   373     qed
```
```   374   qed
```
```   375   thus ?thesis using Node_Node by (auto split:prod.splits)
```
```   376 qed simp_all
```
```   377
```
```   378 text{* Deletion maintains the AVL property: *}
```
```   379
```
```   380 theorem avl_delete_aux:
```
```   381   assumes "avl t"
```
```   382   shows "avl(delete x t)" and "height t = (height (delete x t)) \<or> height t = height (delete x t) + 1"
```
```   383 using assms
```
```   384 proof (induct t)
```
```   385   case (Node h l n r)
```
```   386   case 1
```
```   387   with Node show ?case
```
```   388   proof(cases "x = n")
```
```   389     case True
```
```   390     with Node 1 show ?thesis by (auto simp:avl_delete_root)
```
```   391   next
```
```   392     case False
```
```   393     with Node 1 show ?thesis
```
```   394     proof(cases "x<n")
```
```   395       case True
```
```   396       with Node 1 show ?thesis by (auto simp add:avl_node_bal_r)
```
```   397     next
```
```   398       case False
```
```   399       with Node 1 `x\<noteq>n` show ?thesis by (auto simp add:avl_node_bal_l)
```
```   400     qed
```
```   401   qed
```
```   402   case 2
```
```   403   with Node show ?case
```
```   404   proof(cases "x = n")
```
```   405     case True
```
```   406     with 1 have "height (Node h l n r) = height(delete_root (Node h l n r))
```
```   407       \<or> height (Node h l n r) = height(delete_root (Node h l n r)) + 1"
```
```   408       by (subst height_delete_root,simp_all)
```
```   409     with True show ?thesis by simp
```
```   410   next
```
```   411     case False
```
```   412     with Node 1 show ?thesis
```
```   413      proof(cases "x<n")
```
```   414       case True
```
```   415       show ?thesis
```
```   416       proof(cases "height r = height (delete x l) + 2")
```
```   417         case False with Node 1 `x < n` show ?thesis by(auto simp: node_bal_r_def)
```
```   418       next
```
```   419         case True
```
```   420         hence "(height (node_bal_r (delete x l) n r) = height (delete x l) + 2) \<or>
```
```   421           height (node_bal_r (delete x l) n r) = height (delete x l) + 3" (is "?A \<or> ?B")
```
```   422           using Node 2 by (intro height_node_bal_r) auto
```
```   423         thus ?thesis
```
```   424         proof
```
```   425           assume ?A
```
```   426           with `x < n` Node 2 show ?thesis by(auto simp: node_bal_r_def)
```
```   427         next
```
```   428           assume ?B
```
```   429           with `x < n` Node 2 show ?thesis by(auto simp: node_bal_r_def)
```
```   430         qed
```
```   431       qed
```
```   432     next
```
```   433       case False
```
```   434       show ?thesis
```
```   435       proof(cases "height l = height (delete x r) + 2")
```
```   436         case False with Node 1 `\<not>x < n` `x \<noteq> n` show ?thesis by(auto simp: node_bal_l_def)
```
```   437       next
```
```   438         case True
```
```   439         hence "(height (node_bal_l l n (delete x r)) = height (delete x r) + 2) \<or>
```
```   440           height (node_bal_l l n (delete x r)) = height (delete x r) + 3" (is "?A \<or> ?B")
```
```   441           using Node 2 by (intro height_node_bal_l) auto
```
```   442         thus ?thesis
```
```   443         proof
```
```   444           assume ?A
```
```   445           with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: node_bal_l_def)
```
```   446         next
```
```   447           assume ?B
```
```   448           with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: node_bal_l_def)
```
```   449         qed
```
```   450       qed
```
```   451     qed
```
```   452   qed
```
```   453 qed simp_all
```
```   454
```
```   455 end
```