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.
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 _).
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)).
Global Instance:
Collection A C.
Proof.
split; [split | | |].
* 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.
* intros. unfold intersection_with, listset_nodup_intersection_with,
elem_of, listset_nodup_elem_of. simpl.
rewrite elem_of_remove_dups.
by apply elem_of_list_intersection_with.
Qed.
Global Instance listset_nodup_elems:
Elements A C :=
listset_nodup_car.
Global Instance:
FinCollection A C.
Proof.
split.
* apply _.
* intros. apply elem_of_list_filter.
* done.
* by intros [??].
Qed.
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.