src/HOL/Data_Structures/Tree_Set.thy
 author nipkow Sun Nov 15 12:45:28 2015 +0100 (2015-11-15) changeset 61678 b594e9277be3 parent 61651 415e816d3f37 child 63411 e051eea34990 permissions -rw-r--r--
tuned white space
```     1 (* Author: Tobias Nipkow *)
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```     2
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```     3 section {* Tree Implementation of Sets *}
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```     4
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```     5 theory Tree_Set
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```     6 imports
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```     7   "~~/src/HOL/Library/Tree"
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```     8   Cmp
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```     9   Set_by_Ordered
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```    10 begin
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```    11
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```    12 fun isin :: "'a::cmp tree \<Rightarrow> 'a \<Rightarrow> bool" where
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```    13 "isin Leaf x = False" |
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```    14 "isin (Node l a r) x =
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```    15   (case cmp x a of
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```    16      LT \<Rightarrow> isin l x |
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```    17      EQ \<Rightarrow> True |
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```    18      GT \<Rightarrow> isin r x)"
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```    19
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```    20 hide_const (open) insert
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```    21
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```    22 fun insert :: "'a::cmp \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
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```    23 "insert x Leaf = Node Leaf x Leaf" |
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```    24 "insert x (Node l a r) =
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```    25   (case cmp x a of
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```    26      LT \<Rightarrow> Node (insert x l) a r |
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```    27      EQ \<Rightarrow> Node l a r |
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```    28      GT \<Rightarrow> Node l a (insert x r))"
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```    29
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```    30 fun del_min :: "'a tree \<Rightarrow> 'a * 'a tree" where
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```    31 "del_min (Node l a r) =
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```    32   (if l = Leaf then (a,r) else let (x,l') = del_min l in (x, Node l' a r))"
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```    33
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```    34 fun delete :: "'a::cmp \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
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```    35 "delete x Leaf = Leaf" |
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```    36 "delete x (Node l a r) =
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```    37   (case cmp x a of
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```    38      LT \<Rightarrow>  Node (delete x l) a r |
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```    39      GT \<Rightarrow>  Node l a (delete x r) |
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```    40      EQ \<Rightarrow> if r = Leaf then l else let (a',r') = del_min r in Node l a' r')"
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```    41
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```    42
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```    43 subsection "Functional Correctness Proofs"
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```    44
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```    45 lemma "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
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```    46 by (induction t) (auto simp: elems_simps1)
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```    47
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```    48 lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
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```    49 by (induction t) (auto simp: elems_simps2)
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```    50
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```    51
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```    52 lemma inorder_insert:
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```    53   "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
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```    54 by(induction t) (auto simp: ins_list_simps)
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```    55
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```    56
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```    57 lemma del_minD:
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```    58   "del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> x # inorder t' = inorder t"
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```    59 by(induction t arbitrary: t' rule: del_min.induct)
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```    60   (auto simp: sorted_lems split: prod.splits if_splits)
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```    61
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```    62 lemma inorder_delete:
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```    63   "sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
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```    64 by(induction t) (auto simp: del_list_simps del_minD split: prod.splits)
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```    65
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```    66 interpretation Set_by_Ordered
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```    67 where empty = Leaf and isin = isin and insert = insert and delete = delete
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```    68 and inorder = inorder and inv = "\<lambda>_. True"
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```    69 proof (standard, goal_cases)
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```    70   case 1 show ?case by simp
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```    71 next
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```    72   case 2 thus ?case by(simp add: isin_set)
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```    73 next
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```    74   case 3 thus ?case by(simp add: inorder_insert)
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```    75 next
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```    76   case 4 thus ?case by(simp add: inorder_delete)
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```    77 qed (rule TrueI)+
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```    78
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```    79 end
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