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 ForallOrdPairs_ForallPairs : (forall x, R x x) -> (forall x y, R x y -> R y x) -> forall l, ForallOrdPairs l -> ForallPairs l.