(** [list_disjoint l1 l2] holds iff [l1] and [l2] have no elements
  in common. *)

Definition list_disjoint (A: Type) (l1 l2: list A) : Prop :=
  forall (x y: A), In x l1 -> In y l2 -> x <> y.

Lemma list_disjoint_cons_l: forall (A: Type) (a: A) (l1 l2: list A), list_disjoint l1 l2 -> ~In a l2 -> list_disjoint (a :: l1) l2.