src/HOL/Data_Structures/Tree_Set.thy
author nipkow
Wed Jun 13 15:24:20 2018 +0200 (12 months ago)
changeset 68440 6826718f732d
parent 68431 b294e095f64c
permissions -rw-r--r--
qualify interpretations to avoid clashes
     1 (* Author: Tobias Nipkow *)
     2 
     3 section \<open>Unbalanced Tree Implementation of Set\<close>
     4 
     5 theory Tree_Set
     6 imports
     7   "HOL-Library.Tree"
     8   Cmp
     9   Set_Specs
    10 begin
    11 
    12 definition empty :: "'a tree" where
    13 "empty == Leaf"
    14 
    15 fun isin :: "'a::linorder tree \<Rightarrow> 'a \<Rightarrow> bool" where
    16 "isin Leaf x = False" |
    17 "isin (Node l a r) x =
    18   (case cmp x a of
    19      LT \<Rightarrow> isin l x |
    20      EQ \<Rightarrow> True |
    21      GT \<Rightarrow> isin r x)"
    22 
    23 hide_const (open) insert
    24 
    25 fun insert :: "'a::linorder \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
    26 "insert x Leaf = Node Leaf x Leaf" |
    27 "insert x (Node l a r) =
    28   (case cmp x a of
    29      LT \<Rightarrow> Node (insert x l) a r |
    30      EQ \<Rightarrow> Node l a r |
    31      GT \<Rightarrow> Node l a (insert x r))"
    32 
    33 fun split_min :: "'a tree \<Rightarrow> 'a * 'a tree" where
    34 "split_min (Node l a r) =
    35   (if l = Leaf then (a,r) else let (x,l') = split_min l in (x, Node l' a r))"
    36 
    37 fun delete :: "'a::linorder \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
    38 "delete x Leaf = Leaf" |
    39 "delete x (Node l a r) =
    40   (case cmp x a of
    41      LT \<Rightarrow>  Node (delete x l) a r |
    42      GT \<Rightarrow>  Node l a (delete x r) |
    43      EQ \<Rightarrow> if r = Leaf then l else let (a',r') = split_min r in Node l a' r')"
    44 
    45 
    46 subsection "Functional Correctness Proofs"
    47 
    48 lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> set (inorder t))"
    49 by (induction t) (auto simp: isin_simps)
    50 
    51 lemma inorder_insert:
    52   "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
    53 by(induction t) (auto simp: ins_list_simps)
    54 
    55 
    56 lemma split_minD:
    57   "split_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> x # inorder t' = inorder t"
    58 by(induction t arbitrary: t' rule: split_min.induct)
    59   (auto simp: sorted_lems split: prod.splits if_splits)
    60 
    61 lemma inorder_delete:
    62   "sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
    63 by(induction t) (auto simp: del_list_simps split_minD split: prod.splits)
    64 
    65 interpretation S: Set_by_Ordered
    66 where empty = empty and isin = isin and insert = insert and delete = delete
    67 and inorder = inorder and inv = "\<lambda>_. True"
    68 proof (standard, goal_cases)
    69   case 1 show ?case by (simp add: empty_def)
    70 next
    71   case 2 thus ?case by(simp add: isin_set)
    72 next
    73   case 3 thus ?case by(simp add: inorder_insert)
    74 next
    75   case 4 thus ?case by(simp add: inorder_delete)
    76 qed (rule TrueI)+
    77 
    78 end