intros x' y'; case (Qpositive_le_dec x' y'); auto with *.
case (Qpositive_eq_dec x' y').
intros e q; right; left; rewrite e; auto.
intros n q; right; right; apply Qgt_pos_pos.
red in |- *; intros H; elim n; apply Qpositive_le_antisym; auto.
intros x' y'; case (Qpositive_le_dec x' y'); auto with *.
case (Qpositive_eq_dec x' y').
intros e q; right; left; rewrite e; auto.
intros n q; left; apply Qgt_neg_neg.
red in |- *; intros H; elim n; apply Qpositive_le_antisym; auto.
Qed.
Definition Q_eq_dec : forall x y : Q, {x = y} + {x <> y}.
intros x y; case x; case y; try (right; discriminate).
(* Focus 1 *)
auto.
(* Focus 2 *)
intros y' x'; case (Qpositive_eq_dec x' y').
intros H; left; rewrite H; auto.
intros H; right; red in |- *; intros H'; elim H.
injection H'; auto.
(* Focus 3 *)
intros y' x'; case (Qpositive_eq_dec x' y').
intros H; left; rewrite H; auto.
intros H; right; red in |- *; intros H'; elim H.
injection H'; auto.
Qed.

Theorem Qgt_antisym : forall x y : Q, Qgt x y -> ~ Qgt y x.
intros x y H; elim H.