(** **** Exercise: 2 stars, optional  *)
(** Show that the [total_relation] defined in earlier is not a partial
    function. *)

(* FILL IN HERE *)
(** [] *)

(** **** Exercise: 2 stars, optional  *)
(** Show that the [empty_relation] that we defined earlier is a
    partial function. *)

(* FILL IN HERE *)
(** [] *)

(* ----------------------------------------------------------------- *)
(** *** Reflexive Relations *)

(** A _reflexive_ relation on a set [X] is one for which every element
    of [X] is related to itself. *)

Definition reflexive {X: Type} (R: relation X) :=
  forall a : X, R a a.

(* Specifications: *)
Theorem le_reflexive : reflexive le.