intros L1; elim L1; simpl in |- *; auto.
intros L2 p H' H'0; exists p; split; auto.
split; auto.
apply divp_id; 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 A A0 eqA n ltM
(nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os
a (l ++ L2))); simpl in |- *; auto.
intros Z1.
lapply (nf_divp_zero 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 _ _ _ _ _ _ _ _ _ cs n ltM) with (y := q); auto.
rewrite H'2; auto.
intros Z1.
lapply (nf_divp 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 A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os
a (l ++ L2) :: 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 _ _ _ H'3); auto.
simpl in |- *; auto with datatypes.
simpl in |- *; intros H'6; case H'6;
[ intros H'9; clear H'6 | intros H'9; clear H'6 ];
auto.
auto with datatypes.
apply (divp_trans _ _ _ _ _ _ _ _ _ cs n ltM) with (y := q); auto.
case (in_app_or _ _ _ H'4); auto with datatypes.
rewrite H'2; auto.
case
(zerop_dec A A0 eqA n ltM
(nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os
a (l ++ L2))); simpl in |- *; auto.
intros Z1.
case (in_app_or _ _ _ H'2); auto.
intros H'3.
lapply
(H'
(nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os
a (l ++ L2) :: L2) p);
[ intros H'6; lapply H'6; [ intros H'7; clear H'6 | clear H'6 ] | idtac ];
auto with datatypes.
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 _ _ _ H'4); auto with datatypes.
simpl in |- *; auto.
intros H; case H; intros H1; auto with datatypes.
intros H; exists p; split; [ right | idtac ]; auto with datatypes.
split; auto.
apply divp_id; auto.
Qed.

Theorem Red_divp :
forall (L : list (poly A0 eqA ltM)) (p : poly A0 eqA ltM),
In p L ->
~ zerop A A0 eqA n ltM p ->
exists q : poly A0 eqA ltM,
In q (Red L) /\
divp A A0 eqA multA divA n ltM p q /\ ~ zerop A A0 eqA n ltM q.