intros x y z'; case y.
(* Focus 1 *)
repeat rewrite <- (Qplus_sym Zero).
repeat rewrite Qplus_zero_left; auto with *.
(* Focus 2 *)
case x.
(* 2.1 *)
simpl in |- *; auto with *.
(* 2.2 *)
intros x' y'; simpl in |- *.
rewrite Qpositive_mult_distr.
auto.
(* 2.3 *)
intros x' y'; simpl in |- *.
Case' (Qpositive_le_dec x' y').
case (Qpositive_eq_dec x' y').
simpl in |- *.
intros Heq; rewrite Heq.
Case' (Qpositive_le_dec (Qpositive_mult y' z') (Qpositive_mult y' z')).
case (Qpositive_eq_dec (Qpositive_mult y' z') (Qpositive_mult y' z')).
auto.
intros H; elim H; auto.
intros H; elim H; auto.
Case' (Qpositive_le_dec (Qpositive_mult x' z') (Qpositive_mult y' z')).
case (Qpositive_eq_dec (Qpositive_mult x' z') (Qpositive_mult y' z')).
intros e q n q0; elim n.
apply Qpositive_mult_simpl with z'.
auto.
simpl in |- *; intros; rewrite Qpositive_mult_minus_distr; auto with *.
intros H H0 H1 n q; elim H.
apply Qpositive_le_antisym; auto.
apply Qpositive_le_mult; auto with *.
Case' (Qpositive_le_dec (Qpositive_mult x' z') (Qpositive_mult y' z')).
case (Qpositive_eq_dec (Qpositive_mult x' z') (Qpositive_mult y' z')).
intros e q H H0 H1; elim H.
apply Qpositive_mult_simpl with z'; auto.
intros n q H H0 H1; elim H.
apply Qpositive_mult_simpl with z'; apply Qpositive_le_antisym; auto with *.
apply Qpositive_le_mult; auto.
intros H H0 H3 H1 H2 H4.
simpl in |- *.
rewrite Qpositive_mult_minus_distr; auto.
(* Focus 3 *)
intros y'; case x.
(* 3.1 *)
simpl in |- *; repeat rewrite Qplus_zero_left; auto.
(* 3.2 *)
intros x'; simpl in |- *.
Case' (Qpositive_le_dec x' y').
case (Qpositive_eq_dec x' y').
Case' (Qpositive_le_dec (Qpositive_mult x' z') (Qpositive_mult y' z')).
case (Qpositive_eq_dec (Qpositive_mult x' z') (Qpositive_mult y' z')).
auto.
intros n q e q0; elim n; rewrite e; auto.
intros n H0 H1 e q; elim n; rewrite e; auto.
Case' (Qpositive_le_dec (Qpositive_mult x' z') (Qpositive_mult y' z')).
case (Qpositive_eq_dec (Qpositive_mult x' z') (Qpositive_mult y' z')).
intros e q n; elim n; apply Qpositive_mult_simpl with z'; auto.
simpl in |- *.
intros; rewrite Qpositive_mult_minus_distr; auto with *.
intros H H0 H1 n q; elim H1; apply Qpositive_le_mult; auto with *.
Case' (Qpositive_le_dec (Qpositive_mult x' z') (Qpositive_mult y' z')).
case (Qpositive_eq_dec (Qpositive_mult x' z') (Qpositive_mult y' z')).
intros e q n; elim n; apply Qpositive_mult_simpl with z'; auto.
intros n q H H0 H1; elim n.
apply Qpositive_le_antisym; auto.
apply Qpositive_le_mult; auto with *.
simpl in |- *.
intros; rewrite Qpositive_mult_minus_distr; auto.
(* 3.3 *)
intros x'; simpl in |- *; rewrite Qpositive_mult_distr; auto with *.
Qed.

Theorem Qmult_zero : forall x : Q, Qmult Zero x = Zero.
intros x; case x; simpl in |- *; auto.