intros d; elim d; simpl in |- *; auto.
intros a H' H'0; inversion H'0; auto.
intros n H' a b.
case (le_lt_dec (pmon1 (S n) b) (pmon1 (S n) a)); simpl in |- *; auto.
intros l; generalize (H' (pmon2 (S n) a) (pmon2 (S n) b)).
case (div_mon_clean n (pmon2 (S n) a) (pmon2 (S n) b)); simpl in |- *; auto.
intros H'0 b0 H'1 H'2; red in |- *; intros H'3.
lapply H'1; [ intros H'4; apply H'4; clear H'1 | clear H'1 ]; auto.
change
(pmon2 (S n)
(c_n n (pmon1 (S n) a - pmon1 (S n) b + pmon1 (S n) b)
(mult_mon n (div_mon n (pmon2 (S n) a) (pmon2 (S n) b))
(pmon2 (S n) b))) = pmon2 (S n) a :>mon n)
in |- *.
rewrite H'3; auto.
intros H'0 H'1; red in |- *; intros H'2;
absurd (pmon1 (S n) a - pmon1 (S n) b + pmon1 (S n) b = pmon1 (S n) a);
auto.
2: change
(pmon1 (S n)
(c_n n (pmon1 (S n) a - pmon1 (S n) b + pmon1 (S n) b)
(mult_mon n (div_mon n (pmon2 (S n) a) (pmon2 (S n) b))
(pmon2 (S n) b))) = pmon1 (S n) a :>nat)
in |- *.
2: rewrite H'2; auto.
rewrite (minus_lt_0 (pmon1 (S n) b) (pmon1 (S n) a)); simpl in |- *; auto.
red in |- *; intros H'3; absurd (pmon1 (S n) a < pmon1 (S n) b); auto.
apply le_not_lt; auto.
rewrite H'3; auto.
Qed.

Theorem div_clean_dec1 :
forall (d : nat) (a b : mon d),
gb d (div_mon_clean d a b) = true ->
gm d (div_mon_clean d a b) = div_mon d a b /\
mult_mon d (div_mon d a b) b = a.