intros x' y' Hnle; generalize (not_Qpositive_le_Qpositive_le' _ _ Hnle).
intros Hle; generalize (not_Qpositive_le_not_eq _ _ Hnle).
intros Hneq.
case z; simpl in |- *; auto.
(* z=(Qpos z') *)
intros z'; apply Qgt_pos_pos.
red in |- *; intros Hle'; elim Hneq; apply Qpositive_le_antisym.
apply Qpositive_le_plus_simpl with z';
repeat rewrite <- (Qpositive_plus_sym z'); auto with *.
auto.
(* z=(Qneg z') *)
intros z'; Case' (Qpositive_le_dec z' x').
case (Qpositive_eq_dec z' x').
Case' (Qpositive_le_dec z' y').
intros q e q0.
elim Hnle.
rewrite <- e; auto.
auto.
Case' (Qpositive_le_dec z' y').
case (Qpositive_eq_dec z' y').
auto.
intros n q n0 q0.
apply Qgt_pos_pos.
red in |- *; intros H0.
elim Hnle; apply Qpositive_le_sub_simpl_r with z'; auto with *.
auto.
Case' (Qpositive_le_dec z' y').
intros q H0 H1 H2.
elim Hneq; apply Qpositive_le_antisym; auto with *.
apply Qpositive_le_trans with z'; auto.
intros H0 H1 H4 H2 H3 H5.
apply Qgt_neg_neg.
red in |- *; intros H6.
elim Hnle.
apply Qpositive_le_sub_simpl_l with z'; auto.
(*  (x',y':Qpositive)~(Qpositive_le y' x') *)
intros x' y' Hnle; case z; auto.
(* Focus 1 *)
simpl in |- *; auto.
(* Focus 2 *)
intros z'; simpl in |- *.
Case' (Qpositive_le_dec z' x').
case (Qpositive_eq_dec z' x').
Case' (Qpositive_le_dec z' y').
case (Qpositive_eq_dec z' y').
intros e q e0 q0; elim Hnle; rewrite <- e; rewrite <- e0;
apply Qpositive_le_refl.
auto.
intros H0 H1 H2 e q; elim Hnle; rewrite <- e; auto.
Case' (Qpositive_le_dec z' y').
case (Qpositive_eq_dec z' y').
intros e q n q0; elim Hnle; rewrite <- e; try assumption.
intros n q n0 q0.
apply Qgt_neg_neg.
red in |- *; intros H0.
elim Hnle; apply Qpositive_le_sub_simpl_r with z'; auto with *.
intros H0 H1 H2 n q; elim Hnle; apply Qpositive_le_trans with z'; auto.
Case' (Qpositive_le_dec z' y').
case Qpositive_eq_dec; auto with *.
intros H0 H1 H4 H2 H3 H5.
apply Qgt_pos_pos.
red in |- *; intros H6; elim Hnle.
apply Qpositive_le_sub_simpl_l with z'; auto.
(* Focus 3 *)
simpl in |- *.
intros z'; apply Qgt_neg_neg.
red in |- *; intros H0; elim Hnle; apply Qpositive_le_plus_simpl with z'.
repeat rewrite <- (Qpositive_plus_sym z').
auto.
(*  (Qgt x y)->(x':Qpositive)(Qgt (Qplus z (Qpos x')) (Qplus z Zero)) *)
intros x'; case z.
(* Focus 1 *)
simpl in |- *; auto.
(* Focus 2 *)
intros z'; simpl in |- *.
apply Qgt_pos_pos.
red in |- *; intros H0; elim (Qpositive_plus_diff z' x').
apply Qpositive_le_antisym; auto.
apply Qpositive_plus_le.
(* Focus 3 *)
intros z'; simpl in |- *.
Case' (Qpositive_le_dec z' x').
case (Qpositive_eq_dec z' x'); auto with *.
intros H0 H1 H2; apply Qgt_neg_neg.
red in |- *; intros H3; elim (Qpositive_sub_diff z' x'); auto with *.
apply Qpositive_le_antisym; auto.
apply Qpositive_sub_le; auto.
(*   (x',y':Qpositive)(Qgt (Qplus z (Qpos x')) (Qplus z (Qneg y'))) *)
intros x' y'; case z; simpl in |- *; auto with *.
(* Goal 1 is solved with Auto *)
(* Focus 2 *)
intros z'; Case' (Qpositive_le_dec z' y').
case (Qpositive_eq_dec z' y'); auto with *.
intros H0 H1 H2; apply Qgt_pos_pos.
red in |- *; intros H3; elim (Qpositive_plus_diff z' x').
apply Qpositive_le_antisym.
apply Qpositive_plus_le.
apply Qpositive_le_trans with (Qpositive_sub z' y'); auto with *.
apply Qpositive_sub_le; auto.
(* Focus 3 *)
intros z'; Case' (Qpositive_le_dec z' x').
case (Qpositive_eq_dec z' x').
auto.
auto.
intros H0 H1 H2; apply Qgt_neg_neg.
red in |- *; intros H3; elim (Qpositive_plus_diff z' y').
apply Qpositive_le_antisym.
apply Qpositive_plus_le.
apply Qpositive_le_trans with (Qpositive_sub z' x'); auto with *.
apply Qpositive_sub_le; auto.
(* (x':Qpositive)(Qgt (Qplus z Zero) (Qplus z (Qneg x'))) *)
intros x'; case z; simpl in |- *.
(* Focus 1 *)
auto.
(* Focus 2 *)
intros z'; Case' (Qpositive_le_dec z' x').
case (Qpositive_eq_dec z' x').
auto.
auto.
intros H0 H1 H2; apply Qgt_pos_pos.
red in |- *; intros H3; elim (Qpositive_sub_diff z' x'); auto.
apply Qpositive_le_antisym; auto with *.
apply Qpositive_sub_le; auto.
(* Focus 3 *)
intros z'; apply Qgt_neg_neg.
red in |- *; intros H0; elim (Qpositive_plus_diff z' x').
apply Qpositive_le_antisym; auto.
apply Qpositive_plus_le.
Qed.

Theorem Qgt_mult :
forall x y z : Q, Qgt z Zero -> Qgt x y -> Qgt (Qmult z x) (Qmult z y).