src/HOL/Data_Structures/RBT_Set.thy
 author nipkow Tue Sep 22 08:38:25 2015 +0200 (2015-09-22) changeset 61224 759b5299a9f2 child 61231 cc6969542f8d permissions -rw-r--r--
```     1 (* Author: Tobias Nipkow *)
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```     2
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```     3 section \<open>Red-Black Tree Implementation of Sets\<close>
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```     4
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```     5 theory RBT_Set
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```     6 imports
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```     7   RBT
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```     8   Isin2
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```     9 begin
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```    10
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```    11 fun insert :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
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```    12 "insert x Leaf = R Leaf x Leaf" |
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```    13 "insert x (B l a r) =
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```    14   (if x < a then bal (insert x l) a r else
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```    15    if x > a then bal l a (insert x r) else B l a r)" |
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```    16 "insert x (R l a r) =
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```    17   (if x < a then R (insert x l) a r
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```    18    else if x > a then R l a (insert x r) else R l a r)"
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```    19
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```    20 fun delete :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
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```    21 and deleteL :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
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```    22 and deleteR :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
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```    23 where
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```    24 "delete x Leaf = Leaf" |
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```    25 "delete x (Node _ l a r) =
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```    26   (if x < a then deleteL x l a r
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```    27    else if x > a then deleteR x l a r else combine l r)" |
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```    28 "deleteL x (B t1 a t2) b t3 = balL (delete x (B t1 a t2)) b t3" |
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```    29 "deleteL x l a r = R (delete x l) a r" |
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```    30 "deleteR x t1 a (B t2 b t3) = balR t1 a (delete x (B t2 b t3))" |
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```    31 "deleteR x l a r = R l a (delete x r)"
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```    32
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```    33
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```    34 subsection "Functional Correctness Proofs"
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```    35
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```    36 lemma inorder_bal:
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```    37   "inorder(bal l a r) = inorder l @ a # inorder r"
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```    38 by(induction l a r rule: bal.induct) (auto simp: sorted_lems)
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```    39
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```    40 lemma inorder_insert:
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```    41   "sorted(inorder t) \<Longrightarrow> inorder(insert a t) = ins_list a (inorder t)"
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```    42 by(induction a t rule: insert.induct) (auto simp: ins_simps inorder_bal)
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```    43
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```    44 lemma inorder_red: "inorder(red t) = inorder t"
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```    45 by(induction t) (auto simp: sorted_lems)
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```    46
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```    47 lemma inorder_balL:
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```    48   "inorder(balL l a r) = inorder l @ a # inorder r"
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```    49 by(induction l a r rule: balL.induct)
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```    50   (auto simp: sorted_lems inorder_bal inorder_red)
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```    51
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```    52 lemma inorder_balR:
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```    53   "inorder(balR l a r) = inorder l @ a # inorder r"
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```    54 by(induction l a r rule: balR.induct)
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```    55   (auto simp: sorted_lems inorder_bal inorder_red)
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```    56
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```    57 lemma inorder_combine:
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```    58   "inorder(combine l r) = inorder l @ inorder r"
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```    59 by(induction l r rule: combine.induct)
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```    60   (auto simp: sorted_lems inorder_balL inorder_balR split: tree.split color.split)
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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)" and
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```    64  "sorted(inorder l) \<Longrightarrow>  inorder(deleteL x l a r) =
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```    65     del_list x (inorder l) @ a # inorder r" and
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```    66  "sorted(inorder r) \<Longrightarrow>  inorder(deleteR x l a r) =
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```    67     inorder l @ a # del_list x (inorder r)"
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```    68 by(induction x t and x l a r and x l a r rule: delete_deleteL_deleteR.induct)
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```    69   (auto simp: del_simps inorder_combine inorder_balL inorder_balR)
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```    70
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```    71 interpretation Set_by_Ordered
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```    72 where empty = Leaf and isin = isin and insert = insert and delete = delete
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```    73 and inorder = inorder and wf = "\<lambda>_. True"
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```    74 proof (standard, goal_cases)
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```    75   case 1 show ?case by simp
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```    76 next
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```    77   case 2 thus ?case by(simp add: isin_set)
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```    78 next
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```    79   case 3 thus ?case by(simp add: inorder_insert)
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```    80 next
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```    81   case 4 thus ?case by(simp add: inorder_delete)
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```    82 next
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```    83   case 5 thus ?case ..
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```    84 qed
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```    85
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```    86 end
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