61640
|
1 |
(* Author: Tobias Nipkow *)
|
|
2 |
|
|
3 |
section {* Implementing Ordered Maps *}
|
|
4 |
|
|
5 |
theory Map_by_Ordered
|
|
6 |
imports AList_Upd_Del
|
|
7 |
begin
|
|
8 |
|
|
9 |
locale Map =
|
|
10 |
fixes empty :: "'m"
|
|
11 |
fixes update :: "'a \<Rightarrow> 'b \<Rightarrow> 'm \<Rightarrow> 'm"
|
|
12 |
fixes delete :: "'a \<Rightarrow> 'm \<Rightarrow> 'm"
|
62196
|
13 |
fixes lookup :: "'m \<Rightarrow> 'a \<Rightarrow> 'b option"
|
61640
|
14 |
fixes invar :: "'m \<Rightarrow> bool"
|
62196
|
15 |
assumes map_empty: "lookup empty = (\<lambda>_. None)"
|
|
16 |
and map_update: "invar m \<Longrightarrow> lookup(update a b m) = (lookup m)(a := Some b)"
|
|
17 |
and map_delete: "invar m \<Longrightarrow> lookup(delete a m) = (lookup m)(a := None)"
|
61686
|
18 |
and invar_empty: "invar empty"
|
|
19 |
and invar_update: "invar m \<Longrightarrow> invar(update a b m)"
|
|
20 |
and invar_delete: "invar m \<Longrightarrow> invar(delete a m)"
|
61640
|
21 |
|
|
22 |
locale Map_by_Ordered =
|
|
23 |
fixes empty :: "'t"
|
|
24 |
fixes update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> 't \<Rightarrow> 't"
|
|
25 |
fixes delete :: "'a \<Rightarrow> 't \<Rightarrow> 't"
|
|
26 |
fixes lookup :: "'t \<Rightarrow> 'a \<Rightarrow> 'b option"
|
|
27 |
fixes inorder :: "'t \<Rightarrow> ('a * 'b) list"
|
61686
|
28 |
fixes inv :: "'t \<Rightarrow> bool"
|
61640
|
29 |
assumes empty: "inorder empty = []"
|
61686
|
30 |
and lookup: "inv t \<and> sorted1 (inorder t) \<Longrightarrow>
|
61640
|
31 |
lookup t a = map_of (inorder t) a"
|
61686
|
32 |
and update: "inv t \<and> sorted1 (inorder t) \<Longrightarrow>
|
61640
|
33 |
inorder(update a b t) = upd_list a b (inorder t)"
|
61686
|
34 |
and delete: "inv t \<and> sorted1 (inorder t) \<Longrightarrow>
|
61640
|
35 |
inorder(delete a t) = del_list a (inorder t)"
|
61686
|
36 |
and inv_empty: "inv empty"
|
61688
|
37 |
and inv_update: "inv t \<and> sorted1 (inorder t) \<Longrightarrow> inv(update a b t)"
|
61686
|
38 |
and inv_delete: "inv t \<and> sorted1 (inorder t) \<Longrightarrow> inv(delete a t)"
|
61640
|
39 |
begin
|
|
40 |
|
|
41 |
sublocale Map
|
61688
|
42 |
empty update delete lookup "\<lambda>t. inv t \<and> sorted1 (inorder t)"
|
61640
|
43 |
proof(standard, goal_cases)
|
61688
|
44 |
case 1 show ?case by (auto simp: lookup empty inv_empty)
|
61640
|
45 |
next
|
61688
|
46 |
case 2 thus ?case
|
|
47 |
by(simp add: fun_eq_iff update inv_update map_of_ins_list lookup sorted_upd_list)
|
61640
|
48 |
next
|
61688
|
49 |
case 3 thus ?case
|
|
50 |
by(simp add: fun_eq_iff delete inv_delete map_of_del_list lookup sorted_del_list)
|
61640
|
51 |
next
|
61686
|
52 |
case 4 thus ?case by(simp add: empty inv_empty)
|
61640
|
53 |
next
|
61688
|
54 |
case 5 thus ?case by(simp add: update inv_update sorted_upd_list)
|
61640
|
55 |
next
|
61686
|
56 |
case 6 thus ?case by (auto simp: delete inv_delete sorted_del_list)
|
61640
|
57 |
qed
|
|
58 |
|
|
59 |
end
|
|
60 |
|
|
61 |
end
|