Variables (x y: e).
Definition Elt: Prop := Ecmp _ x y = Lt.
Definition Egt: Prop := Ecmp _ x y = Gt.
Definition Ele: Prop := Ecmp _ x y <> Gt.
Definition Ege: Prop := Ecmp _ x y <> Lt.
Definition Eltb: bool := match Ecmp _ x y with Lt => true | _ => false end.
Definition Egtb: bool := match Ecmp _ x y with Gt => true | _ => false end.
Definition Eleb: bool := match Ecmp _ x y with Gt => false | _ => true end.
Definition Egeb: bool := match Ecmp _ x y with Lt => false | _ => true end.

(* Specifications: *)
Lemma Eltb_true: Elt <-> Eltb = true.
Lemma Ele_dec x y: decision (Ele x y).
Lemma Ecmp_refl x: Ecmp e x x = Eq.
Proof with auto.