(** [] *)

(** A relation [R] is _antisymmetric_ if [R a b] and [R b a] together
    imply [a = b] -- that is, if the only "cycles" in [R] are trivial
    ones. *)

Definition antisymmetric {X: Type} (R: relation X) :=
  forall a b : X, (R a b) -> (R b a) -> a = b.

(* Specifications: *)
Theorem le_antisymmetric : antisymmetric le.
Theorem le_step : forall n m p, n < m -> m <= S p -> n <= p.