src/HOL/Data_Structures/Tree_Map.thy
 author nipkow Sun Apr 08 11:05:52 2018 +0200 (15 months ago) changeset 67965 aaa31cd0caef parent 63411 e051eea34990 child 68020 6aade817bee5 permissions -rw-r--r--
more name tuning
```     1 (* Author: Tobias Nipkow *)
```
```     2
```
```     3 section \<open>Unbalanced Tree Implementation of Map\<close>
```
```     4
```
```     5 theory Tree_Map
```
```     6 imports
```
```     7   Tree_Set
```
```     8   Map_Specs
```
```     9 begin
```
```    10
```
```    11 fun lookup :: "('a::linorder*'b) tree \<Rightarrow> 'a \<Rightarrow> 'b option" where
```
```    12 "lookup Leaf x = None" |
```
```    13 "lookup (Node l (a,b) r) x =
```
```    14   (case cmp x a of LT \<Rightarrow> lookup l x | GT \<Rightarrow> lookup r x | EQ \<Rightarrow> Some b)"
```
```    15
```
```    16 fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
```
```    17 "update x y Leaf = Node Leaf (x,y) Leaf" |
```
```    18 "update x y (Node l (a,b) r) = (case cmp x a of
```
```    19    LT \<Rightarrow> Node (update x y l) (a,b) r |
```
```    20    EQ \<Rightarrow> Node l (x,y) r |
```
```    21    GT \<Rightarrow> Node l (a,b) (update x y r))"
```
```    22
```
```    23 fun delete :: "'a::linorder \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
```
```    24 "delete x Leaf = Leaf" |
```
```    25 "delete x (Node l (a,b) r) = (case cmp x a of
```
```    26   LT \<Rightarrow> Node (delete x l) (a,b) r |
```
```    27   GT \<Rightarrow> Node l (a,b) (delete x r) |
```
```    28   EQ \<Rightarrow> if r = Leaf then l else let (ab',r') = del_min r in Node l ab' r')"
```
```    29
```
```    30
```
```    31 subsection "Functional Correctness Proofs"
```
```    32
```
```    33 lemma lookup_map_of:
```
```    34   "sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
```
```    35 by (induction t) (auto simp: map_of_simps split: option.split)
```
```    36
```
```    37 lemma inorder_update:
```
```    38   "sorted1(inorder t) \<Longrightarrow> inorder(update a b t) = upd_list a b (inorder t)"
```
```    39 by(induction t) (auto simp: upd_list_simps)
```
```    40
```
```    41 lemma inorder_delete:
```
```    42   "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
```
```    43 by(induction t) (auto simp: del_list_simps del_minD split: prod.splits)
```
```    44
```
```    45 interpretation Map_by_Ordered
```
```    46 where empty = Leaf and lookup = lookup and update = update and delete = delete
```
```    47 and inorder = inorder and inv = "\<lambda>_. True"
```
```    48 proof (standard, goal_cases)
```
```    49   case 1 show ?case by simp
```
```    50 next
```
```    51   case 2 thus ?case by(simp add: lookup_map_of)
```
```    52 next
```
```    53   case 3 thus ?case by(simp add: inorder_update)
```
```    54 next
```
```    55   case 4 thus ?case by(simp add: inorder_delete)
```
```    56 qed auto
```
```    57
```
```    58 end
```