assert (Ecmp e x x = CompOpp (Ecmp e x x)).
apply Ecmp_sym.
destruct (Ecmp e x x); try discriminate...
Qed.

(* Specifications: *)
Lemma Ele_le_dec x y: { Ele x y } + { Ele y x }.
Lemma Ecmp_inv_sym x y c: Ecmp e x y <> CompOpp c -> Ecmp e y x <> c.
Lemma Ele_Ege x y: Ele x y -> Ege y x.
Lemma Ecmp_lt_le_trans: forall x y z, Ecmp e x y = Lt -> Ele y z -> Ecmp e x z = Lt.
Proof with auto.