src/HOL/Data_Structures/Tree_Map.thy
 author nipkow Thu Nov 05 08:27:14 2015 +0100 (2015-11-05) changeset 61581 00d9682e8dd7 parent 61534 a88e07c8d0d5 child 61640 44c9198f210c permissions -rw-r--r--
Convertd to 3-way comparisons
```     1 (* Author: Tobias Nipkow *)
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```     2
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```     3 section {* Unbalanced Tree as Map *}
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```     4
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```     5 theory Tree_Map
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```     6 imports
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```     7   Tree_Set
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```     8   Map_by_Ordered
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```     9 begin
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```    10
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```    11 fun lookup :: "('a::cmp*'b) tree \<Rightarrow> 'a \<Rightarrow> 'b option" where
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```    12 "lookup Leaf x = None" |
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```    13 "lookup (Node l (a,b) r) x =
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```    14   (case cmp x a of LT \<Rightarrow> lookup l x | GT \<Rightarrow> lookup r x | EQ \<Rightarrow> Some b)"
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```    15
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```    16 fun update :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
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```    17 "update x y Leaf = Node Leaf (x,y) Leaf" |
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```    18 "update x y (Node l (a,b) r) = (case cmp x a of
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```    19    LT \<Rightarrow> Node (update x y l) (a,b) r |
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```    20    EQ \<Rightarrow> Node l (x,y) r |
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```    21    GT \<Rightarrow> Node l (a,b) (update x y r))"
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```    22
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```    23 fun delete :: "'a::cmp \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
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```    24 "delete x Leaf = Leaf" |
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```    25 "delete x (Node l (a,b) r) = (case cmp x a of
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```    26   LT \<Rightarrow> Node (delete x l) (a,b) r |
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```    27   GT \<Rightarrow> Node l (a,b) (delete x r) |
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```    28   EQ \<Rightarrow> if r = Leaf then l else let (ab',r') = del_min r in Node l ab' r')"
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```    29
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```    30
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```    31 subsection "Functional Correctness Proofs"
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```    32
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```    33 lemma lookup_eq:
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```    34   "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
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```    35 by (induction t) (auto simp: map_of_simps split: option.split)
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```    36
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```    37
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```    38 lemma inorder_update:
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```    39   "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
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```    40 by(induction t) (auto simp: upd_list_simps)
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```    41
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```    42
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```    43 lemma del_minD:
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```    44   "del_min t = (x,t') \<Longrightarrow> t \<noteq> Leaf \<Longrightarrow> sorted1(inorder t) \<Longrightarrow>
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```    45    x # inorder t' = inorder t"
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```    46 by(induction t arbitrary: t' rule: del_min.induct)
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```    47   (auto simp: del_list_simps split: prod.splits)
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```    48
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```    49 lemma inorder_delete:
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```    50   "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
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```    51 by(induction t) (auto simp: del_list_simps del_minD split: prod.splits)
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```    52
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```    53 interpretation Map_by_Ordered
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```    54 where empty = Leaf and lookup = lookup and update = update and delete = delete
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```    55 and inorder = inorder and wf = "\<lambda>_. True"
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```    56 proof (standard, goal_cases)
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```    57   case 1 show ?case by simp
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```    58 next
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```    59   case 2 thus ?case by(simp add: lookup_eq)
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```    60 next
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```    61   case 3 thus ?case by(simp add: inorder_update)
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```    62 next
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```    63   case 4 thus ?case by(simp add: inorder_delete)
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```    64 qed (rule TrueI)+
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```    65
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```    66 end
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