simpl in |- *; intros n' m' p'.
Case' (Qpositive_le_dec m' p').
intros Hle; case (Qpositive_eq_dec m' p').
intros Heq; Case' (Qpositive_le_dec (Qpositive_plus n' m') p').
intros Hle1.
elim (Qpositive_plus_diff p' n').
apply Qpositive_le_antisym.
apply Qpositive_plus_le.
generalize Hle1; rewrite Heq; rewrite Qpositive_plus_sym; auto.
intros H H0 H1; rewrite <- Heq.
rewrite Qpositive_sub_correct; auto.
Case' (Qpositive_le_dec n' (Qpositive_sub p' m')).
case (Qpositive_eq_dec n' (Qpositive_sub p' m')).
intros e q n0; Case' (Qpositive_le_dec (Qpositive_plus n' m') p').
case (Qpositive_eq_dec (Qpositive_plus n' m') p').
auto.
intros n1 q0.
elim n1; rewrite e.
rewrite Qpositive_sub_correct2; auto.
intros Hnot1 Hle' Hnot; elim Hnot; rewrite e.
rewrite Qpositive_sub_correct2; auto.
apply Qpositive_le_refl.
Case' (Qpositive_le_dec (Qpositive_plus n' m') p').
case (Qpositive_eq_dec (Qpositive_plus n' m') p').
intros e q n0 q0 n1.
elim n0.
rewrite <- e.
rewrite Qpositive_sub_correct; auto.
intros e q n0 q0 n1.
rewrite Qpositive_sub_sub_assoc_l; auto.
rewrite Qpositive_plus_sym; auto.
rewrite Qpositive_plus_sym; auto.
rewrite <- (Qpositive_sub_correct2 p' m'); auto.
rewrite (Qpositive_plus_sym m'); apply Qpositive_le_add; auto with *.
intros H H0 H1 n0 q n1.
elim n0; apply Qpositive_le_antisym; auto.
rewrite <- (Qpositive_sub_correct n' m').
apply Qpositive_le_sub_r; auto.
rewrite Qpositive_plus_sym; apply sym_not_equal; apply Qpositive_plus_diff.
rewrite Qpositive_plus_sym; apply Qpositive_plus_le.
Case' (Qpositive_le_dec (Qpositive_plus n' m') p').
case (Qpositive_eq_dec (Qpositive_plus n' m') p').
intros e q H H0 H1 n0.
elim H1; rewrite <- e; rewrite Qpositive_sub_correct; apply Qpositive_le_refl.
intros n0 q H H0 H1 n1.
elim H1.
rewrite <- (Qpositive_sub_correct n' m').
apply Qpositive_le_sub_r; auto with *.
apply sym_not_equal; rewrite Qpositive_plus_sym; apply Qpositive_plus_diff.
rewrite Qpositive_plus_sym; apply Qpositive_plus_le.
intros H H0 H3 H1 H2 H4 n0.
rewrite Qpositive_sub_sub_assoc; auto with *.
Case' (Qpositive_le_dec (Qpositive_plus n' m') p').
case (Qpositive_eq_dec (Qpositive_plus n' m') p').
intros e q H H0 H1.
elim H1.
rewrite <- e; rewrite Qpositive_plus_sym; apply Qpositive_plus_le.
intros n0 q H H0 H1.
elim H1.
apply Qpositive_le_trans with (Qpositive_plus n' m'); auto.
rewrite Qpositive_plus_sym; apply Qpositive_plus_le.
intros H H0 H3 H1 H2 H4.
rewrite Qpositive_plus_sym.
rewrite Qpositive_plus_sub_assoc; auto with *.
rewrite Qpositive_plus_sym; auto.
Qed.

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