intros n; elim n; clear n.
intros n' p q p' q' d; rewrite <- plus_n_Sm.
simpl in |- *.
intros H1 H2 H3; inversion H3.
inversion H0.
intros n0 Hrec n' p q p' q' d Heqd1 Heqd2 Hle1 Hle2.
simpl in |- *.
CaseEq (p - q).
CaseEq (q - p).
intros H H0.
cut (p' = q').
intros Heq'; rewrite Heq'; rewrite <- (minus_n_n q').
auto.
apply mult_reg_l with d.
rewrite <- Heqd2; rewrite <- Heqd1.
apply le_antisym; apply minus_O_le; assumption.
intros q2 Heq2 Heqp.
cut (p' <= q').
intros H; generalize (le_minus_O _ _ H).
intros Heq3; rewrite Heq3.
CaseEq (q' - p').
intros Heq4; cut (q' = p').
intros Heq5; generalize Heq2; replace (q - p) with (S q - S p).
rewrite Heqd2; rewrite Heqd1; rewrite Heq5.
rewrite <- minus_n_n; intros Heq6; discriminate Heq6.
auto.
apply le_antisym; apply minus_O_le; assumption.
intros q'2.
generalize Hle2; case n'.
generalize Heqd1 Heqd2; case p'; case q'.
simpl in |- *; intros H'1 H'2 H'3 H'4; discriminate H'4.
intros x; rewrite <- (mult_comm 0); simpl in |- *; intros Dummy;
discriminate Dummy.
simpl in |- *; intros x H'1 H'2 H'3 H'4; discriminate H'4.
simpl in |- *; intros n n1 H'1 H'2; rewrite <- plus_n_Sm; intros H'3;
inversion H'3.
inversion H1.
intros n''.
intros Hle3 Heq4.
simpl in |- *.
CaseEq p'.
intros Heqp'; generalize Heqd1; rewrite Heqp'.
rewrite <- (mult_comm 0); simpl in |- *; intros Dummy; discriminate Dummy.
intros p'2 Heqp'2.
change
(dL (Qpositive_c (S p) (S q2) (S n0)) =
dL (Qpositive_c (S p'2) (S q'2) (S n''))) in |- *.
apply f_equal with (f := dL).
apply Hrec with d; auto with *.
rewrite <- Heqp'2; assumption.
rewrite <- Heq4; rewrite <- Heq2; rewrite (mult_comm (S d)).
rewrite mult_minus_distr_r.
repeat rewrite <- (mult_comm (S d)).
rewrite <- Heqd2; rewrite <- Heqd1.
simpl in |- *; auto.
apply mult_S_le_reg_l with d.
rewrite <- Heqd2; rewrite <- Heqd1.
apply le_n_S.
apply minus_O_le; assumption.
intros p2 Heqp2.
CaseEq (p' - q').
intros Heqm'; cut (S p - S q = 0).
simpl in |- *; rewrite Heqp2; intros Dummy; discriminate Dummy.
rewrite Heqd2; rewrite Heqd1.
repeat rewrite (mult_comm (S d)).
rewrite <- mult_minus_distr_r.
rewrite Heqm'; simpl in |- *; auto.
intros p'2.
generalize Hle2; case n'.
generalize Heqd1 Heqd2; case p'; case q'.
simpl in |- *; intros H'1 H'2 H'3 H'4; discriminate H'4.
intros x; rewrite <- (mult_comm 0); simpl in |- *; intros Dummy;
discriminate Dummy.
intros x; rewrite <- (mult_comm 0); simpl in |- *; intros Dummy1 Dummy2;
discriminate Dummy2.
simpl in |- *; intros n n1 H'1 H'2; rewrite <- plus_n_Sm; intros H'3;
inversion H'3.
inversion H0.
intros n''.
intros Hle3 Heq4.
CaseEq q'.
intros Heq5; generalize Heqd2; rewrite Heq5; simpl in |- *.
rewrite <- (mult_comm 0); simpl in |- *; intros Dummy; discriminate Dummy.
intros q'2 Heqq'2.
change
(nR (Qpositive_c (S p2) (S q) (S n0)) =
nR (Qpositive_c (S p'2) (S q'2) (S n''))) in |- *.
apply f_equal with (f := nR).
apply Hrec with d; auto with *.
rewrite <- Heq4; rewrite <- Heqp2; rewrite (mult_comm (S d)).
rewrite mult_minus_distr_r.
repeat rewrite <- (mult_comm (S d)).
rewrite <- Heqd2; rewrite <- Heqd1.
simpl in |- *; auto.
rewrite <- Heqq'2; assumption.
Qed.

Theorem construct_correct4 :
forall p q p' q' n n' : nat,
S p + S q <= S n ->
S p' + S q' <= S n' ->
S p * S q' = S p' * S q ->
Qpositive_c (S p) (S q) (S n) = Qpositive_c (S p') (S q') (S n').