intros n; case n.
(* Focus 1 *)
intros; repeat rewrite Qplus_zero_left; auto.
(* Focus 2 *)
simpl in |- *; intros n' m' p'.
Case' (Qpositive_le_dec n' (Qpositive_plus m' p')).
case (Qpositive_eq_dec n' (Qpositive_plus m' p')).
Case' (Qpositive_le_dec n' m').
case (Qpositive_eq_dec n' m').
intros e q e0; elim (Qpositive_plus_diff n' p').
pattern n' at 2 in |- *; rewrite e; auto.
intros n0 q e q0; elim n0; apply Qpositive_le_antisym; auto.
rewrite e; apply Qpositive_plus_le.
intros H H0 H1 e q; simpl in |- *.
Case' (Qpositive_le_dec (Qpositive_sub n' m') p').
case (Qpositive_eq_dec (Qpositive_sub n' m') p'); auto with *.
rewrite e; intros H2; elim H2.
rewrite Qpositive_plus_sym; rewrite Qpositive_sub_correct; auto.
rewrite e; intros H2; elim H2.
rewrite Qpositive_plus_sym; rewrite Qpositive_sub_correct; auto.
Case' (Qpositive_le_dec n' m').
case (Qpositive_eq_dec n' m').
simpl in |- *.
rewrite Qpositive_plus_sym.
intros e q n0 q0; rewrite e; rewrite Qpositive_sub_correct; auto with *.
simpl in |- *.
intros; rewrite Qpositive_plus_sub_assoc; auto.
simpl in |- *.
Case' (Qpositive_le_dec (Qpositive_sub n' m') p').
case (Qpositive_eq_dec (Qpositive_sub n' m') p').
intros e q H H0 H1 n0 q0; elim n0.
symmetry  in |- *; apply add_sub_eq2; auto with *.
intros; rewrite Qpositive_sub_sub_assoc; auto.
rewrite Qpositive_plus_sym; auto.
intros H H0 H3 H1 H2 H4 n0 q; elim H3.
apply add_sub_le2; auto.
Case' (Qpositive_le_dec n' m').
case (Qpositive_eq_dec n' m').
intros e q H H0 H1; elim H1; rewrite e; apply Qpositive_plus_le.
simpl in |- *.
intros n0 q H H0 H1; elim H1; apply Qpositive_le_trans with m'; auto with *.
apply Qpositive_plus_le.
simpl in |- *.
Case' (Qpositive_le_dec (Qpositive_sub n' m') p').
case (Qpositive_eq_dec (Qpositive_sub n' m') p').
intros e q H H0 H2 H1; elim H1.
symmetry  in |- *; apply add_sub_eq2; auto with *.
intros n0 q H H0 H3 H1 H2 H4; elim H4; apply add_sub_le1; auto with *.
intros; rewrite Qpositive_sub_sub_assoc_l; auto with *.
(* Focus 3 *)
intros m' n' p'; simpl in |- *.
rewrite Qpositive_plus_assoc; auto with *.
Qed.

Theorem Qplus_assoc_one_neg :
forall n' m' p' : Qpositive,
Qplus (Qpos n') (Qplus (Qpos m') (Qneg p')) =
Qplus (Qplus (Qpos n') (Qpos m')) (Qneg p').