src/HOL/Data_Structures/RBT_Set.thy
author nipkow
Tue Oct 13 17:06:37 2015 +0200 (2015-10-13)
changeset 61428 5e1938107371
parent 61231 cc6969542f8d
child 61581 00d9682e8dd7
permissions -rw-r--r--
added invar empty
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(* Author: Tobias Nipkow *)
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section \<open>Red-Black Tree Implementation of Sets\<close>
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theory RBT_Set
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imports
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  RBT
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  Isin2
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begin
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fun insert :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
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"insert x Leaf = R Leaf x Leaf" |
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"insert x (B l a r) =
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  (if x < a then bal (insert x l) a r else
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   if x > a then bal l a (insert x r) else B l a r)" |
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"insert x (R l a r) =
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  (if x < a then R (insert x l) a r
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   else if x > a then R l a (insert x r) else R l a r)"
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fun delete :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
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and deleteL :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
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and deleteR :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
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where
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"delete x Leaf = Leaf" |
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"delete x (Node _ l a r) = 
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  (if x < a then deleteL x l a r 
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   else if x > a then deleteR x l a r else combine l r)" |
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"deleteL x (B t1 a t2) b t3 = balL (delete x (B t1 a t2)) b t3" |
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"deleteL x l a r = R (delete x l) a r" |
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"deleteR x t1 a (B t2 b t3) = balR t1 a (delete x (B t2 b t3))" | 
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"deleteR x l a r = R l a (delete x r)"
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subsection "Functional Correctness Proofs"
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lemma inorder_bal:
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  "inorder(bal l a r) = inorder l @ a # inorder r"
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by(induction l a r rule: bal.induct) (auto)
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lemma inorder_insert:
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  "sorted(inorder t) \<Longrightarrow> inorder(insert a t) = ins_list a (inorder t)"
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by(induction a t rule: insert.induct) (auto simp: ins_list_simps inorder_bal)
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lemma inorder_red: "inorder(red t) = inorder t"
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by(induction t) (auto)
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lemma inorder_balL:
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  "inorder(balL l a r) = inorder l @ a # inorder r"
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by(induction l a r rule: balL.induct)(auto simp: inorder_bal inorder_red)
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lemma inorder_balR:
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  "inorder(balR l a r) = inorder l @ a # inorder r"
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by(induction l a r rule: balR.induct) (auto simp: inorder_bal inorder_red)
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lemma inorder_combine:
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  "inorder(combine l r) = inorder l @ inorder r"
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by(induction l r rule: combine.induct)
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  (auto simp: inorder_balL inorder_balR split: tree.split color.split)
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lemma inorder_delete:
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 "sorted(inorder t) \<Longrightarrow>  inorder(delete x t) = del_list x (inorder t)" and
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 "sorted(inorder l) \<Longrightarrow>  inorder(deleteL x l a r) =
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    del_list x (inorder l) @ a # inorder r" and
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 "sorted(inorder r) \<Longrightarrow>  inorder(deleteR x l a r) =
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    inorder l @ a # del_list x (inorder r)"
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by(induction x t and x l a r and x l a r rule: delete_deleteL_deleteR.induct)
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  (auto simp: del_list_simps inorder_combine inorder_balL inorder_balR)
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interpretation Set_by_Ordered
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where empty = Leaf and isin = isin and insert = insert and delete = delete
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and inorder = inorder and wf = "\<lambda>_. True"
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proof (standard, goal_cases)
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  case 1 show ?case by simp
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next
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  case 2 thus ?case by(simp add: isin_set)
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next
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  case 3 thus ?case by(simp add: inorder_insert)
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next
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  case 4 thus ?case by(simp add: inorder_delete)
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qed (rule TrueI)+
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end