Library stdpp.listset_nodup

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.
From stdpp Require Export sets list.
From stdpp Require Import options.

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

Section list_set.
Context `{EqDecision A}.
Notation C := (listset_nodup A).

Global Instance listset_nodup_elem_of: ElemOf A C := λ x l, x listset_nodup_car l.
Global Instance listset_nodup_empty: Empty C := ListsetNoDup [] (@NoDup_nil_2 _).
Global Instance listset_nodup_singleton: Singleton A C := λ x,
  ListsetNoDup [x] (NoDup_singleton x).
Global Instance listset_nodup_union: Union C :=
  λ '(ListsetNoDup l Hl) '(ListsetNoDup k Hk),
    ListsetNoDup _ (NoDup_list_union _ _ Hl Hk).
Global Instance listset_nodup_intersection: Intersection C :=
  λ '(ListsetNoDup l Hl) '(ListsetNoDup k Hk),
    ListsetNoDup _ (NoDup_list_intersection _ k Hl).
Global Instance listset_nodup_difference: Difference C :=
  λ '(ListsetNoDup l Hl) '(ListsetNoDup k Hk),
    ListsetNoDup _ (NoDup_list_difference _ k Hl).

Instance listset_nodup_set: Set_ 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 listset_nodup_fin_set: FinSet A C.
Proof. split; [apply _|done|]. by intros [??]. Qed.
End list_set.