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 NoDup_disjointness_1: forall {A} (l1 l2 : list A), NoDup l1 -> NoDup l2 -> disjoint_list l1 l2 -> NoDup (l1 ++ l2).