Qed.
Inductive Qgt : Q -> Q -> Prop :=
| Qgt_pos_pos :
forall x' y' : Qpositive,
~ Qpositive_le x' y' -> Qgt (Qpos x') (Qpos y')
| Qgt_neg_neg :
forall x' y' : Qpositive,
~ Qpositive_le y' x' -> Qgt (Qneg x') (Qneg y')
| Qgt_pos_zero : forall x' : Qpositive, Qgt (Qpos x') Zero
| Qgt_pos_neg : forall x' y' : Qpositive, Qgt (Qpos x') (Qneg y')
| Qgt_zero_neg : forall x' : Qpositive, Qgt Zero (Qneg x').
Hint Resolve Qgt_pos_pos Qgt_neg_neg Qgt_pos_zero Qgt_pos_neg Qgt_zero_neg.

Theorem Qgt_total : forall x y : Q, Qgt x y \/ x = y \/ Qgt y x.
intros x y; case y; case x; auto.