intros n; elim n.
simpl in |- *; intros p q n' p' q' H; inversion H.
intros n0 Hrec p q n'.
case n'.
simpl in |- *; intros p' q' H H1; inversion H1.
unfold Qpositive_c in |- *.
CaseEq (S p - S q).
CaseEq (S q - S p).
intros Heq1 Heq2 n1 p' q'; CaseEq (S p' - S q').
CaseEq (S q' - S p').
intros Heq3 Heq4; cut (p = q).
intros Heq5; rewrite Heq5.
cut (p' = q').
intros Heq6; rewrite Heq6.
intros; apply mult_comm.
apply eq_add_S; apply le_antisym; apply minus_O_le; auto with *.
apply eq_add_S; apply le_antisym; apply minus_O_le; auto with *.
intros p2 H H1 H2 H3 H4; discriminate H4.
intros n2 H H1 H2 H4; discriminate H4.
intros q2 Heq1 Heq2 n2 p' q'; CaseEq (S p' - S q').
CaseEq (S q' - S p').
intros H3 H4 H H1 H2; discriminate H2.
intros q'2 H H1 H2 H3 H4; injection H4; intros H5.
rewrite (le_plus_minus (S p) (S q)).
rewrite (le_plus_minus (S p') (S q')).
rewrite (mult_comm (S p)); rewrite (mult_comm (S p')).
repeat rewrite mult_plus_distr_r; repeat rewrite <- (mult_comm (S p)).
apply f_equal with (f := fun x : nat => S p * S p' + x).
rewrite Heq1; rewrite H.
rewrite <- (mult_comm (S p')).
apply Hrec with n2.
rewrite <- Heq1.
rewrite <- le_plus_minus.
apply le_trans with (p + S q).
auto with arith.
apply le_S_n; exact H2.
apply minus_O_le; auto.
rewrite <- H.
rewrite <- le_plus_minus.
apply le_trans with (p' + S q').
auto with arith; fail.
apply le_S_n; exact H3.
apply minus_O_le; auto.
exact H5.
apply minus_O_le; auto.
apply minus_O_le; auto.
intros n1 dummy1 Dummy2 Dummy3 Dummy; discriminate Dummy.
intros n1 Heq1 n2 p' q' Hle1 Hle2; CaseEq (S p' - S q').
CaseEq (S q' - S p').
intros Dummy3 Dummy1 Dummy2; discriminate Dummy2.
intros n3 Dummy Dummy1 Dummy2; discriminate Dummy2.
intros p'2 Heq2 Heq3.
injection Heq3; clear Heq3; intros Heq3.
rewrite (le_plus_minus (S q) (S p)).
rewrite Heq1.
rewrite (le_plus_minus (S q') (S p')).
rewrite Heq2.
repeat rewrite mult_plus_distr_r; repeat rewrite (mult_comm (S q')).
apply f_equal with (f := fun x : nat => S q * S q' + x).
apply Hrec with n2.
rewrite plus_comm; rewrite <- Heq1.
rewrite <- le_plus_minus.
apply le_trans with (p + S q).
rewrite <- plus_n_Sm.
auto with arith.
apply le_S_n; assumption.
omega.
rewrite <- Heq2.
rewrite plus_comm; rewrite <- le_plus_minus.
apply le_trans with (p' + S q').
rewrite <- plus_n_Sm; auto with arith.
apply le_S_n; exact Hle2.
omega.
exact Heq3.
omega.
omega.
Qed.

Theorem Qpositive_c_equiv' :
forall n p q n' p' q' : nat,
1 <= p ->
1 <= q ->
1 <= p' ->
1 <= q' ->
p + q <= n ->
p' + q' <= n' -> Qpositive_c p q n = Qpositive_c p' q' n' -> p * q' = p' * q.