Definition ForallPairs l :=
    forall a b, In a l -> In b l -> R a b.(** [ForallOrdPairs] : we still check a relation over all pairs
     of elements of a list, but now the order of elements matters. *)

  Inductive ForallOrdPairs : list A -> Prop :=
    | FOP_nil : ForallOrdPairs []
    | FOP_cons : forall a l,
      Forall (R a) l -> ForallOrdPairs l -> ForallOrdPairs (a::l).(** [ForallOrdPairs] : we still check a relation over all pairs
     of elements of a list, but now the order of elements matters. *)

  Inductive ForallOrdPairs : list A -> Prop :=
    | FOP_nil : ForallOrdPairs []
    | FOP_cons : forall a l,
      Forall (R a) l -> ForallOrdPairs l -> ForallOrdPairs (a::l).(** [ForallOrdPairs] : we still check a relation over all pairs
     of elements of a list, but now the order of elements matters. *)

  Inductive ForallOrdPairs : list A -> Prop :=
    | FOP_nil : ForallOrdPairs []
    | FOP_cons : forall a l,
      Forall (R a) l -> ForallOrdPairs l -> ForallOrdPairs (a::l).

Lemma NoDup_concat [A] (L: list (list A)): Forall (@NoDup A) L -> ForallOrdPairs (fun l1 l2 => forall a, In a l1 -> ~ In a l2) L -> NoDup (concat L).