src/HOL/Data_Structures/RBT_Set.thy
 author nipkow Fri Nov 27 18:01:13 2015 +0100 (2015-11-27) changeset 61749 7f530d7e552d parent 61678 b594e9277be3 child 61754 862daa8144f3 permissions -rw-r--r--
paint root black after insert and delete
```     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   Cmp
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```     9   Isin2
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```    10 begin
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```    11
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```    12 fun ins :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
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```    13 "ins x Leaf = R Leaf x Leaf" |
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```    14 "ins x (B l a r) =
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```    15   (case cmp x a of
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```    16      LT \<Rightarrow> bal (ins x l) a r |
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```    17      GT \<Rightarrow> bal l a (ins x r) |
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```    18      EQ \<Rightarrow> B l a r)" |
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```    19 "ins x (R l a r) =
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```    20   (case cmp x a of
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```    21     LT \<Rightarrow> R (ins x l) a r |
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```    22     GT \<Rightarrow> R l a (ins x r) |
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```    23     EQ \<Rightarrow> R l a r)"
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```    24
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```    25 definition insert :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
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```    26 "insert x t = paint Black (ins x t)"
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```    27
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```    28 fun del :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
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```    29 and delL :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
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```    30 and delR :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
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```    31 where
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```    32 "del x Leaf = Leaf" |
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```    33 "del x (Node _ l a r) =
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```    34   (case cmp x a of
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```    35      LT \<Rightarrow> delL x l a r |
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```    36      GT \<Rightarrow> delR x l a r |
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```    37      EQ \<Rightarrow> combine l r)" |
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```    38 "delL x (B t1 a t2) b t3 = balL (del x (B t1 a t2)) b t3" |
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```    39 "delL x l a r = R (del x l) a r" |
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```    40 "delR x t1 a (B t2 b t3) = balR t1 a (del x (B t2 b t3))" |
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```    41 "delR x l a r = R l a (del x r)"
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```    42
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```    43 definition delete :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
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```    44 "delete x t = paint Black (del x t)"
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```    45
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```    46
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```    47 subsection "Functional Correctness Proofs"
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```    48
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```    49 lemma inorder_paint: "inorder(paint c t) = inorder t"
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```    50 by(induction t) (auto)
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```    51
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```    52 lemma inorder_bal:
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```    53   "inorder(bal l a r) = inorder l @ a # inorder r"
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```    54 by(induction l a r rule: bal.induct) (auto)
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```    55
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```    56 lemma inorder_ins:
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```    57   "sorted(inorder t) \<Longrightarrow> inorder(ins x t) = ins_list x (inorder t)"
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```    58 by(induction x t rule: ins.induct) (auto simp: ins_list_simps inorder_bal)
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```    59
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```    60 lemma inorder_insert:
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```    61   "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
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```    62 by (simp add: insert_def inorder_ins inorder_paint)
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```    63
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```    64 lemma inorder_balL:
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```    65   "inorder(balL l a r) = inorder l @ a # inorder r"
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```    66 by(induction l a r rule: balL.induct)(auto simp: inorder_bal inorder_paint)
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```    67
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```    68 lemma inorder_balR:
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```    69   "inorder(balR l a r) = inorder l @ a # inorder r"
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```    70 by(induction l a r rule: balR.induct) (auto simp: inorder_bal inorder_paint)
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```    71
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```    72 lemma inorder_combine:
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```    73   "inorder(combine l r) = inorder l @ inorder r"
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```    74 by(induction l r rule: combine.induct)
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```    75   (auto simp: inorder_balL inorder_balR split: tree.split color.split)
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```    76
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```    77 lemma inorder_del:
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```    78  "sorted(inorder t) \<Longrightarrow>  inorder(del x t) = del_list x (inorder t)"
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```    79  "sorted(inorder l) \<Longrightarrow>  inorder(delL x l a r) =
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```    80     del_list x (inorder l) @ a # inorder r"
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```    81  "sorted(inorder r) \<Longrightarrow>  inorder(delR x l a r) =
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```    82     inorder l @ a # del_list x (inorder r)"
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```    83 by(induction x t and x l a r and x l a r rule: del_delL_delR.induct)
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```    84   (auto simp: del_list_simps inorder_combine inorder_balL inorder_balR)
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```    85
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```    86 lemma inorder_delete:
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```    87   "sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
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```    88 by (auto simp: delete_def inorder_del inorder_paint)
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```    89
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```    90
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```    91 interpretation Set_by_Ordered
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```    92 where empty = Leaf and isin = isin and insert = insert and delete = delete
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```    93 and inorder = inorder and inv = "\<lambda>_. True"
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```    94 proof (standard, goal_cases)
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```    95   case 1 show ?case by simp
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```    96 next
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```    97   case 2 thus ?case by(simp add: isin_set)
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```    98 next
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```    99   case 3 thus ?case by(simp add: inorder_insert)
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```   100 next
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```   101   case 4 thus ?case by(simp add: inorder_delete)
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```   102 qed auto
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```   103
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```   104 end
```