src/HOL/Data_Structures/RBT_Set.thy
author nipkow
Thu Nov 05 08:27:14 2015 +0100 (2015-11-05)
changeset 61581 00d9682e8dd7
parent 61428 5e1938107371
child 61588 1d2907d0ed73
permissions -rw-r--r--
Convertd to 3-way comparisons
nipkow@61224
     1
(* Author: Tobias Nipkow *)
nipkow@61224
     2
nipkow@61224
     3
section \<open>Red-Black Tree Implementation of Sets\<close>
nipkow@61224
     4
nipkow@61224
     5
theory RBT_Set
nipkow@61224
     6
imports
nipkow@61224
     7
  RBT
nipkow@61581
     8
  Cmp
nipkow@61224
     9
  Isin2
nipkow@61224
    10
begin
nipkow@61224
    11
nipkow@61581
    12
fun insert :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
nipkow@61224
    13
"insert x Leaf = R Leaf x Leaf" |
nipkow@61581
    14
"insert x (B l a r) = (case cmp x a of
nipkow@61581
    15
  LT \<Rightarrow> bal (insert x l) a r |
nipkow@61581
    16
  GT \<Rightarrow> bal l a (insert x r) |
nipkow@61581
    17
  EQ \<Rightarrow> B l a r)" |
nipkow@61581
    18
"insert x (R l a r) = (case cmp x a of
nipkow@61581
    19
  LT \<Rightarrow> R (insert x l) a r |
nipkow@61581
    20
  GT \<Rightarrow> R l a (insert x r) |
nipkow@61581
    21
  EQ \<Rightarrow> R l a r)"
nipkow@61224
    22
nipkow@61581
    23
fun delete :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
nipkow@61581
    24
and deleteL :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
nipkow@61581
    25
and deleteR :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
nipkow@61224
    26
where
nipkow@61224
    27
"delete x Leaf = Leaf" |
nipkow@61581
    28
"delete x (Node _ l a r) = (case cmp x a of
nipkow@61581
    29
  LT \<Rightarrow> deleteL x l a r |
nipkow@61581
    30
  GT \<Rightarrow> deleteR x l a r |
nipkow@61581
    31
  EQ \<Rightarrow> combine l r)" |
nipkow@61224
    32
"deleteL x (B t1 a t2) b t3 = balL (delete x (B t1 a t2)) b t3" |
nipkow@61224
    33
"deleteL x l a r = R (delete x l) a r" |
nipkow@61224
    34
"deleteR x t1 a (B t2 b t3) = balR t1 a (delete x (B t2 b t3))" | 
nipkow@61224
    35
"deleteR x l a r = R l a (delete x r)"
nipkow@61224
    36
nipkow@61224
    37
nipkow@61224
    38
subsection "Functional Correctness Proofs"
nipkow@61224
    39
nipkow@61224
    40
lemma inorder_bal:
nipkow@61224
    41
  "inorder(bal l a r) = inorder l @ a # inorder r"
nipkow@61231
    42
by(induction l a r rule: bal.induct) (auto)
nipkow@61224
    43
nipkow@61224
    44
lemma inorder_insert:
nipkow@61224
    45
  "sorted(inorder t) \<Longrightarrow> inorder(insert a t) = ins_list a (inorder t)"
nipkow@61231
    46
by(induction a t rule: insert.induct) (auto simp: ins_list_simps inorder_bal)
nipkow@61224
    47
nipkow@61224
    48
lemma inorder_red: "inorder(red t) = inorder t"
nipkow@61231
    49
by(induction t) (auto)
nipkow@61224
    50
nipkow@61224
    51
lemma inorder_balL:
nipkow@61224
    52
  "inorder(balL l a r) = inorder l @ a # inorder r"
nipkow@61231
    53
by(induction l a r rule: balL.induct)(auto simp: inorder_bal inorder_red)
nipkow@61224
    54
nipkow@61224
    55
lemma inorder_balR:
nipkow@61224
    56
  "inorder(balR l a r) = inorder l @ a # inorder r"
nipkow@61231
    57
by(induction l a r rule: balR.induct) (auto simp: inorder_bal inorder_red)
nipkow@61224
    58
nipkow@61224
    59
lemma inorder_combine:
nipkow@61224
    60
  "inorder(combine l r) = inorder l @ inorder r"
nipkow@61224
    61
by(induction l r rule: combine.induct)
nipkow@61231
    62
  (auto simp: inorder_balL inorder_balR split: tree.split color.split)
nipkow@61224
    63
nipkow@61224
    64
lemma inorder_delete:
nipkow@61224
    65
 "sorted(inorder t) \<Longrightarrow>  inorder(delete x t) = del_list x (inorder t)" and
nipkow@61224
    66
 "sorted(inorder l) \<Longrightarrow>  inorder(deleteL x l a r) =
nipkow@61224
    67
    del_list x (inorder l) @ a # inorder r" and
nipkow@61224
    68
 "sorted(inorder r) \<Longrightarrow>  inorder(deleteR x l a r) =
nipkow@61224
    69
    inorder l @ a # del_list x (inorder r)"
nipkow@61224
    70
by(induction x t and x l a r and x l a r rule: delete_deleteL_deleteR.induct)
nipkow@61231
    71
  (auto simp: del_list_simps inorder_combine inorder_balL inorder_balR)
nipkow@61224
    72
nipkow@61581
    73
nipkow@61224
    74
interpretation Set_by_Ordered
nipkow@61224
    75
where empty = Leaf and isin = isin and insert = insert and delete = delete
nipkow@61224
    76
and inorder = inorder and wf = "\<lambda>_. True"
nipkow@61224
    77
proof (standard, goal_cases)
nipkow@61224
    78
  case 1 show ?case by simp
nipkow@61224
    79
next
nipkow@61224
    80
  case 2 thus ?case by(simp add: isin_set)
nipkow@61224
    81
next
nipkow@61224
    82
  case 3 thus ?case by(simp add: inorder_insert)
nipkow@61224
    83
next
nipkow@61224
    84
  case 4 thus ?case by(simp add: inorder_delete)
nipkow@61428
    85
qed (rule TrueI)+
nipkow@61224
    86
nipkow@61224
    87
end