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.

Lemma preorder_impl X (P Q: relation X): (forall x y, P x y <-> Q x y) -> preorder _ P -> preorder _ Q.