listset_nodup.v 3.78 KB
Newer Older
1
(* Copyright (c) 2012-2013, Robbert Krebbers. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
(* This file is distributed under the terms of the BSD license. *)
(** This file implements finite as unordered lists without duplicates.
Although this implementation is slow, it is very useful as decidable equality
is the only constraint on the carrier set. *)
Require Export base decidable collections list.

Record listset_nodup A := ListsetNoDup {
  listset_nodup_car : list A;
  listset_nodup_prf : NoDup listset_nodup_car
}.
Arguments ListsetNoDup {_} _ _.
Arguments listset_nodup_car {_} _.
Arguments listset_nodup_prf {_} _.

Section list_collection.
Context {A : Type} `{ x y : A, Decision (x = y)}.

Notation C := (listset_nodup A).
Notation LS := ListsetNoDup.

22
23
Instance listset_nodup_elem_of: ElemOf A C := λ x l, x  listset_nodup_car l.
Instance listset_nodup_empty: Empty C := LS [] (@NoDup_nil_2 _).
Robbert Krebbers's avatar
Robbert Krebbers committed
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
Instance listset_nodup_singleton: Singleton A C := λ x,
  LS [x] (NoDup_singleton x).
Instance listset_nodup_difference: Difference C := λ l k,
  LS _ (list_difference_nodup _ (listset_nodup_car k) (listset_nodup_prf l)).

Definition listset_nodup_union_raw (l k : list A) : list A :=
  list_difference l k ++ k.
Lemma elem_of_listset_nodup_union_raw l k x :
  x  listset_nodup_union_raw l k  x  l  x  k.
Proof.
  unfold listset_nodup_union_raw.
  rewrite elem_of_app, elem_of_list_difference.
  intuition. case (decide (x  k)); intuition.
Qed.
Lemma listset_nodup_union_raw_nodup l k :
  NoDup l  NoDup k  NoDup (listset_nodup_union_raw l k).
Proof.
  intros. apply NoDup_app. repeat split.
  * by apply list_difference_nodup.
  * intro. rewrite elem_of_list_difference. intuition.
  * done.
Qed.
Instance listset_nodup_union: Union C := λ l k,
  LS _ (listset_nodup_union_raw_nodup _ _
     (listset_nodup_prf l) (listset_nodup_prf k)).
Instance listset_nodup_intersection: Intersection C := λ l k,
  LS _ (list_intersection_nodup _
     (listset_nodup_car k) (listset_nodup_prf l)).
Instance listset_nodup_intersection_with:
    IntersectionWith A C := λ f l k,
  LS (remove_dups
      (list_intersection_with f (listset_nodup_car l) (listset_nodup_car k)))
    (remove_dups_nodup _).
Instance listset_nodup_filter: Filter A C :=
  λ P _ l, LS _ (filter_nodup P _ (listset_nodup_prf l)).

60
Instance: Collection A C.
Robbert Krebbers's avatar
Robbert Krebbers committed
61
Proof.
62
  split; [split | | ].
Robbert Krebbers's avatar
Robbert Krebbers committed
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
  * by apply not_elem_of_nil.
  * by apply elem_of_list_singleton.
  * intros. apply elem_of_listset_nodup_union_raw.
  * intros. apply elem_of_list_intersection.
  * intros. apply elem_of_list_difference.
Qed.

Global Instance listset_nodup_elems: Elements A C := listset_nodup_car.

Global Instance: FinCollection A C.
Proof.
  split.
  * apply _.
  * done.
  * by intros [??].
Qed.
79
80
81
82
83
84
85

Global Instance: CollectionOps A C.
Proof.
  split.
  * apply _.
  * intros. unfold intersection_with, listset_nodup_intersection_with,
      elem_of, listset_nodup_elem_of. simpl.
86
    rewrite elem_of_remove_dups. by apply elem_of_list_intersection_with.
87
88
  * intros. apply elem_of_list_filter.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
End list_collection.

Hint Extern 1 (ElemOf _ (listset_nodup _)) =>
  eapply @listset_nodup_elem_of : typeclass_instances.
Hint Extern 1 (Empty (listset_nodup _)) =>
  eapply @listset_nodup_empty : typeclass_instances.
Hint Extern 1 (Singleton _ (listset_nodup _)) =>
  eapply @listset_nodup_singleton : typeclass_instances.
Hint Extern 1 (Union (listset_nodup _)) =>
  eapply @listset_nodup_union : typeclass_instances.
Hint Extern 1 (Intersection (listset_nodup _)) =>
  eapply @listset_nodup_intersection : typeclass_instances.
Hint Extern 1 (Difference (listset_nodup _)) =>
  eapply @listset_nodup_difference : typeclass_instances.
Hint Extern 1 (Elements _ (listset_nodup _)) =>
  eapply @listset_nodup_elems : typeclass_instances.
Hint Extern 1 (Filter _ (listset_nodup _)) =>
  eapply @listset_nodup_filter : typeclass_instances.