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).
Instance listset_nodup_elem_of:
ElemOf A C := λ
x l,
x ∈
listset_nodup_car l.
Instance listset_nodup_empty:
Empty C :=
ListsetNoDup [] (@
NoDup_nil_2 _).
Instance listset_nodup_singleton:
Singleton A C := λ
x,
ListsetNoDup [
x] (
NoDup_singleton x).
Instance listset_nodup_union:
Union C := λ
l k,
let (
l',
Hl) :=
l in let (
k',
Hk) :=
k
in ListsetNoDup _ (
NoDup_list_union _ _ Hl Hk).
Instance listset_nodup_intersection:
Intersection C := λ
l k,
let (
l',
Hl) :=
l in let (
k',
Hk) :=
k
in ListsetNoDup _ (
NoDup_list_intersection _ k'
Hl).
Instance listset_nodup_difference:
Difference C := λ
l k,
let (
l',
Hl) :=
l in let (
k',
Hk) :=
k
in ListsetNoDup _ (
NoDup_list_difference _ k'
Hl).
Instance listset_nodup_intersection_with:
IntersectionWith A C := λ
f l k,
let (
l',
Hl) :=
l in let (
k',
Hk) :=
k
in ListsetNoDup
(
remove_dups (
list_intersection_with f l'
k')) (
NoDup_remove_dups _).
Instance listset_nodup_filter:
Filter A C := λ
P _ l,
let (
l',
Hl) :=
l in ListsetNoDup _ (
NoDup_filter P _ Hl).
Instance:
Collection A C.
Proof.
split; [split | | ].
* by apply not_elem_of_nil.
* by apply elem_of_list_singleton.
* intros [??] [??] ?. apply elem_of_list_union.
* 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.
Global Instance:
CollectionOps A C.
Proof.
split.
* apply _.
* intros ? [??] [??] ?. unfold intersection_with, elem_of,
listset_nodup_intersection_with, listset_nodup_elem_of; simpl.
rewrite elem_of_remove_dups. by apply elem_of_list_intersection_with.
* intros [??] ???. apply elem_of_list_filter.
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.