Definition compare x y : Compare lt eq x y.Definition eq_dec := eq_nat_dec.Definition t := Z.Definition eq := @eq Z.Definition eq_refl := @eq_refl t.Definition eq_sym := @eq_sym t.Definition eq_trans := @eq_trans t.Definition lt (x y:Z) := (x<y).

Lemma lt_not_eq : forall x y, x<y -> ~ x=y.