src/HOL/Library/Tree.thy
 author wenzelm Wed Jun 17 11:03:05 2015 +0200 (2015-06-17) changeset 60500 903bb1495239 parent 59928 b9b7f913a19a child 60506 83231b558ce4 permissions -rw-r--r--
isabelle update_cartouches;
```     1 (* Author: Tobias Nipkow *)
```
```     2
```
```     3 section \<open>Binary Tree\<close>
```
```     4
```
```     5 theory Tree
```
```     6 imports Main
```
```     7 begin
```
```     8
```
```     9 datatype 'a tree =
```
```    10   Leaf ("\<langle>\<rangle>") |
```
```    11   Node (left: "'a tree") (val: 'a) (right: "'a tree") ("\<langle>_, _, _\<rangle>")
```
```    12   where
```
```    13     "left Leaf = Leaf"
```
```    14   | "right Leaf = Leaf"
```
```    15 datatype_compat tree
```
```    16
```
```    17 text\<open>Can be seen as counting the number of leaves rather than nodes:\<close>
```
```    18
```
```    19 definition size1 :: "'a tree \<Rightarrow> nat" where
```
```    20 "size1 t = size t + 1"
```
```    21
```
```    22 lemma size1_simps[simp]:
```
```    23   "size1 \<langle>\<rangle> = 1"
```
```    24   "size1 \<langle>l, x, r\<rangle> = size1 l + size1 r"
```
```    25 by (simp_all add: size1_def)
```
```    26
```
```    27 lemma neq_Leaf_iff: "(t \<noteq> \<langle>\<rangle>) = (\<exists>l a r. t = \<langle>l, a, r\<rangle>)"
```
```    28 by (cases t) auto
```
```    29
```
```    30 lemma finite_set_tree[simp]: "finite(set_tree t)"
```
```    31 by(induction t) auto
```
```    32
```
```    33 lemma size_map_tree[simp]: "size (map_tree f t) = size t"
```
```    34 by (induction t) auto
```
```    35
```
```    36 lemma size1_map_tree[simp]: "size1 (map_tree f t) = size1 t"
```
```    37 by (simp add: size1_def)
```
```    38
```
```    39
```
```    40 subsection "The depth"
```
```    41
```
```    42 fun depth :: "'a tree => nat" where
```
```    43 "depth Leaf = 0" |
```
```    44 "depth (Node t1 a t2) = Suc (max (depth t1) (depth t2))"
```
```    45
```
```    46 lemma depth_map_tree[simp]: "depth (map_tree f t) = depth t"
```
```    47 by (induction t) auto
```
```    48
```
```    49
```
```    50 subsection "The set of subtrees"
```
```    51
```
```    52 fun subtrees :: "'a tree \<Rightarrow> 'a tree set" where
```
```    53   "subtrees \<langle>\<rangle> = {\<langle>\<rangle>}" |
```
```    54   "subtrees (\<langle>l, a, r\<rangle>) = insert \<langle>l, a, r\<rangle> (subtrees l \<union> subtrees r)"
```
```    55
```
```    56 lemma set_treeE: "a \<in> set_tree t \<Longrightarrow> \<exists>l r. \<langle>l, a, r\<rangle> \<in> subtrees t"
```
```    57 by (induction t)(auto)
```
```    58
```
```    59 lemma Node_notin_subtrees_if[simp]: "a \<notin> set_tree t \<Longrightarrow> Node l a r \<notin> subtrees t"
```
```    60 by (induction t) auto
```
```    61
```
```    62 lemma in_set_tree_if: "\<langle>l, a, r\<rangle> \<in> subtrees t \<Longrightarrow> a \<in> set_tree t"
```
```    63 by (metis Node_notin_subtrees_if)
```
```    64
```
```    65
```
```    66 subsection "List of entries"
```
```    67
```
```    68 fun preorder :: "'a tree \<Rightarrow> 'a list" where
```
```    69 "preorder \<langle>\<rangle> = []" |
```
```    70 "preorder \<langle>l, x, r\<rangle> = x # preorder l @ preorder r"
```
```    71
```
```    72 fun inorder :: "'a tree \<Rightarrow> 'a list" where
```
```    73 "inorder \<langle>\<rangle> = []" |
```
```    74 "inorder \<langle>l, x, r\<rangle> = inorder l @ [x] @ inorder r"
```
```    75
```
```    76 lemma set_inorder[simp]: "set (inorder t) = set_tree t"
```
```    77 by (induction t) auto
```
```    78
```
```    79 lemma set_preorder[simp]: "set (preorder t) = set_tree t"
```
```    80 by (induction t) auto
```
```    81
```
```    82 lemma length_preorder[simp]: "length (preorder t) = size t"
```
```    83 by (induction t) auto
```
```    84
```
```    85 lemma length_inorder[simp]: "length (inorder t) = size t"
```
```    86 by (induction t) auto
```
```    87
```
```    88 lemma preorder_map: "preorder (map_tree f t) = map f (preorder t)"
```
```    89 by (induction t) auto
```
```    90
```
```    91 lemma inorder_map: "inorder (map_tree f t) = map f (inorder t)"
```
```    92 by (induction t) auto
```
```    93
```
```    94
```
```    95 subsection \<open>Binary Search Tree predicate\<close>
```
```    96
```
```    97 fun (in linorder) bst :: "'a tree \<Rightarrow> bool" where
```
```    98 "bst \<langle>\<rangle> \<longleftrightarrow> True" |
```
```    99 "bst \<langle>l, a, r\<rangle> \<longleftrightarrow> bst l \<and> bst r \<and> (\<forall>x\<in>set_tree l. x < a) \<and> (\<forall>x\<in>set_tree r. a < x)"
```
```   100
```
```   101 text\<open>In case there are duplicates:\<close>
```
```   102
```
```   103 fun (in linorder) bst_eq :: "'a tree \<Rightarrow> bool" where
```
```   104 "bst_eq \<langle>\<rangle> \<longleftrightarrow> True" |
```
```   105 "bst_eq \<langle>l,a,r\<rangle> \<longleftrightarrow>
```
```   106  bst_eq l \<and> bst_eq r \<and> (\<forall>x\<in>set_tree l. x \<le> a) \<and> (\<forall>x\<in>set_tree r. a \<le> x)"
```
```   107
```
```   108 lemma (in linorder) bst_eq_if_bst: "bst t \<Longrightarrow> bst_eq t"
```
```   109 by (induction t) (auto)
```
```   110
```
```   111 lemma (in linorder) bst_eq_imp_sorted: "bst_eq t \<Longrightarrow> sorted (inorder t)"
```
```   112 apply (induction t)
```
```   113  apply(simp)
```
```   114 by (fastforce simp: sorted_append sorted_Cons intro: less_imp_le less_trans)
```
```   115
```
```   116 lemma (in linorder) distinct_preorder_if_bst: "bst t \<Longrightarrow> distinct (preorder t)"
```
```   117 apply (induction t)
```
```   118  apply simp
```
```   119 apply(fastforce elim: order.asym)
```
```   120 done
```
```   121
```
```   122 lemma (in linorder) distinct_inorder_if_bst: "bst t \<Longrightarrow> distinct (inorder t)"
```
```   123 apply (induction t)
```
```   124  apply simp
```
```   125 apply(fastforce elim: order.asym)
```
```   126 done
```
```   127
```
```   128
```
```   129 subsection "Function @{text mirror}"
```
```   130
```
```   131 fun mirror :: "'a tree \<Rightarrow> 'a tree" where
```
```   132 "mirror \<langle>\<rangle> = Leaf" |
```
```   133 "mirror \<langle>l,x,r\<rangle> = \<langle>mirror r, x, mirror l\<rangle>"
```
```   134
```
```   135 lemma mirror_Leaf[simp]: "mirror t = \<langle>\<rangle> \<longleftrightarrow> t = \<langle>\<rangle>"
```
```   136 by (induction t) simp_all
```
```   137
```
```   138 lemma size_mirror[simp]: "size(mirror t) = size t"
```
```   139 by (induction t) simp_all
```
```   140
```
```   141 lemma size1_mirror[simp]: "size1(mirror t) = size1 t"
```
```   142 by (simp add: size1_def)
```
```   143
```
```   144 lemma depth_mirror[simp]: "depth(mirror t) = depth t"
```
```   145 by (induction t) simp_all
```
```   146
```
```   147 lemma inorder_mirror: "inorder(mirror t) = rev(inorder t)"
```
```   148 by (induction t) simp_all
```
```   149
```
```   150 lemma map_mirror: "map_tree f (mirror t) = mirror (map_tree f t)"
```
```   151 by (induction t) simp_all
```
```   152
```
```   153 lemma mirror_mirror[simp]: "mirror(mirror t) = t"
```
```   154 by (induction t) simp_all
```
```   155
```
```   156
```
```   157 subsection "Deletion of the rightmost entry"
```
```   158
```
```   159 fun del_rightmost :: "'a tree \<Rightarrow> 'a tree * 'a" where
```
```   160 "del_rightmost \<langle>l, a, \<langle>\<rangle>\<rangle> = (l,a)" |
```
```   161 "del_rightmost \<langle>l, a, r\<rangle> = (let (r',x) = del_rightmost r in (\<langle>l, a, r'\<rangle>, x))"
```
```   162
```
```   163 lemma del_rightmost_set_tree_if_bst:
```
```   164   "\<lbrakk> del_rightmost t = (t',x); bst t; t \<noteq> Leaf \<rbrakk>
```
```   165   \<Longrightarrow> x \<in> set_tree t \<and> set_tree t' = set_tree t - {x}"
```
```   166 apply(induction t arbitrary: t' rule: del_rightmost.induct)
```
```   167   apply (fastforce simp: ball_Un split: prod.splits)+
```
```   168 done
```
```   169
```
```   170 lemma del_rightmost_set_tree:
```
```   171   "\<lbrakk> del_rightmost t = (t',x);  t \<noteq> \<langle>\<rangle> \<rbrakk> \<Longrightarrow> set_tree t = insert x (set_tree t')"
```
```   172 apply(induction t arbitrary: t' rule: del_rightmost.induct)
```
```   173 by (auto split: prod.splits) auto
```
```   174
```
```   175 lemma del_rightmost_bst:
```
```   176   "\<lbrakk> del_rightmost t = (t',x);  bst t;  t \<noteq> \<langle>\<rangle> \<rbrakk> \<Longrightarrow> bst t'"
```
```   177 proof(induction t arbitrary: t' rule: del_rightmost.induct)
```
```   178   case (2 l a rl b rr)
```
```   179   let ?r = "Node rl b rr"
```
```   180   from "2.prems"(1) obtain r' where 1: "del_rightmost ?r = (r',x)" and [simp]: "t' = Node l a r'"
```
```   181     by(simp split: prod.splits)
```
```   182   from "2.prems"(2) 1 del_rightmost_set_tree[OF 1] show ?case by(auto)(simp add: "2.IH")
```
```   183 qed auto
```
```   184
```
```   185
```
```   186 lemma del_rightmost_greater: "\<lbrakk> del_rightmost t = (t',x);  bst t;  t \<noteq> \<langle>\<rangle> \<rbrakk>
```
```   187   \<Longrightarrow> \<forall>a\<in>set_tree t'. a < x"
```
```   188 proof(induction t arbitrary: t' rule: del_rightmost.induct)
```
```   189   case (2 l a rl b rr)
```
```   190   from "2.prems"(1) obtain r'
```
```   191   where dm: "del_rightmost (Node rl b rr) = (r',x)" and [simp]: "t' = Node l a r'"
```
```   192     by(simp split: prod.splits)
```
```   193   show ?case using "2.prems"(2) "2.IH"[OF dm] del_rightmost_set_tree_if_bst[OF dm]
```
```   194     by (fastforce simp add: ball_Un)
```
```   195 qed simp_all
```
```   196
```
```   197 lemma del_rightmost_Max:
```
```   198   "\<lbrakk> del_rightmost t = (t',x);  bst t;  t \<noteq> \<langle>\<rangle> \<rbrakk> \<Longrightarrow> x = Max(set_tree t)"
```
```   199 by (metis Max_insert2 del_rightmost_greater del_rightmost_set_tree finite_set_tree less_le_not_le)
```
```   200
```
```   201 end
```