intros L1; elim L1; simpl in |- *; auto.
intros L2 p H' H'0; exists p; split; auto.
unfold LetP in |- *.
intros a l H' L2 p H'0 H'1; case H'1;
[ intros H'2; rewrite <- H'2; clear H'1 | intros H'2; clear H'1 ];
auto.
case (zerop_dec (nf (l ++ L2) a)); simpl in |- *; auto.
intros Z1.
lapply (nf_div_zero1 a (l ++ L2));
[ intros H'5; lapply H'5; [ intros H'6; clear H'5 | clear H'5 ] | idtac ];
auto.
case H'6; intros q E; case E; intros H'3 H'4; case H'4; intros H'5 H'7;
clear H'4 E H'6.
lapply (H' L2 q);
[ intros H'8; lapply H'8; [ intros H'9; clear H'8 | clear H'8 ] | idtac ];
auto.
case H'9; intros q0 E; case E; intros H'4 H'6; case H'6; intros H'8 H'10;
clear H'6 E H'9.
exists q0; split; [ idtac | split ]; auto.
apply divp_trans with (y := q); auto.
rewrite H'2; auto.
intros Z1.
lapply (nf_div a (l ++ L2));
[ intros H'5; lapply H'5; [ intros H'6; clear H'5 | clear H'5 ] | idtac ];
auto.
case H'6; intros q E; case E; intros H'3 H'4; case H'4; intros H'5 H'7;
clear H'4 E H'6.
simpl in H'3.
case H'3; [ intros H'4; clear H'3 | intros H'4; clear H'3 ].
exists q; split; [ idtac | split ]; auto.
lapply (H' (nf (l ++ L2) a :: L2) q);
[ intros H'8; lapply H'8; [ intros H'9; clear H'8 | clear H'8 ] | idtac ];
auto.
case H'9; intros q0 E; case E; intros H'3 H'6; case H'6; intros H'8 H'10;
clear H'6 E H'9.
exists q0; split; [ idtac | split ]; auto.
case
(in_app_or (redacc l (nf (l ++ L2) a :: L2)) (nf (l ++ L2) a :: L2) q0 H'3);
auto.
simpl in |- *; auto.
intros H'6; case H'6; [ intros H'9; clear H'6 | intros H'9; clear H'6 ]; auto.
apply divp_trans with (y := q); auto.
case (in_app_or l L2 q H'4); auto with datatypes.
rewrite H'2; auto.
case (zerop_dec (nf (l ++ L2) a)); simpl in |- *; auto.
intros Z1.
case (in_app_or l L2 p H'2); auto.
intros H'3.
lapply (H' (nf (l ++ L2) a :: L2) p);
[ intros H'6; lapply H'6; [ intros H'7; clear H'6 | clear H'6 ] | idtac ];
auto.
case H'7; intros q E; case E; intros H'4 H'5; case H'5; intros H'6 H'8;
clear H'5 E H'7.
exists q; split; auto.
case
(in_app_or (redacc l (nf (l ++ L2) a :: L2)) (nf (l ++ L2) a :: L2) q H'4);
auto.
simpl in |- *; auto.
intros H; case H; intros H1; auto.
intros H; exists p; split; [ right | idtac ]; auto.
Qed.

Theorem red_divp :
forall (L : list poly) (p : poly),
In p L ->
~ zerop p -> exists q : poly, In q (red L) /\ divp p q /\ ~ zerop q.