Definition inj_on_list {A B} (l : list A) (f : A -> B) :=
  forall x y, In x l -> In y l -> f x = f y -> x = y.Definition subset_rel {X} (sub lst : list X) : Prop :=
  forall x,
    In x sub ->
    In x lst.

Lemma remove_aux_removed {A} `{EqDec A} : forall (lst : list A) (a : A), NoDup lst -> ~ In a (remove_aux eqb lst a).