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_map_NoDup_ForallPairs [A B] (f: A->B) (l: list A) : ForallPairs (fun x y => f x = f y -> x = y) l -> NoDup l -> NoDup (map f l).