intros aP R Q i H'; elim H'; clear H' i aP R Q; simpl in |- *; auto.
intros i L H' b H'0; elim H'0.
intros L L1 L2 L3 a i H' H'0 H'1 H'2 b H'3 H'4.
cut (incl L2 L1);
[ intros incl0
| generalize (slicef_incl i a L1); unfold slicef in |- *; rewrite H' ];
auto.
elim H'3; [ intros H'5; rewrite <- H'5; clear H'3 | intros H'5; clear H'3 ];
auto.
exists a; split; [ idtac | split; [ left | idtac ] ]; auto.
rewrite H'5; auto.
apply zerop_ddivp_ppc; auto.
elim (slice_inv i a L1 b); [ intros H'10 | intros H'10 | idtac ]; auto.
rewrite H' in H'10; simpl in H'10; auto.
lapply H'1;
[ intros H'3; elim (H'3 b);
[ intros c E; elim E; intros H'9 H'11; elim H'11; intros H'12 H'13;
elim H'12;
[ intros H'14; clear H'12 H'11 E H'1
| intros H'14; clear H'12 H'11 E H'1 ]
| clear H'1
| clear H'1 ]
| clear H'1 ]; auto.
exists c; split; [ right | idtac ]; auto.
exists c; split; [ idtac | split; [ right | idtac ] ]; auto.
exists a; split; [ idtac | split ]; auto.
intros L L1 L2 L3 a i H' H'0 H'1 H'2 b H'3 H'4.
cut (incl L2 L1);
[ intros incl0
| generalize (slicef_incl i a L1); unfold slicef in |- *; rewrite H' ];
auto.
elim H'3; [ intros H'5; rewrite <- H'5; clear H'3 | intros H'5; clear H'3 ];
auto.
elim (slice_cons i a L1 L2);
[ intros H'8; elim H'8; intros c E; elim E; intros H'9 H'10; clear E H'8
| intros H'8
| idtac ]; auto.
lapply H'1;
[ intros H'3; elim (H'3 c);
[ intros c0 E; elim E; intros H'11 H'12; elim H'12; intros H'13 H'14;
elim H'13;
[ intros H'15; clear H'13 H'12 E H'1
| intros H'15; clear H'13 H'12 E H'1 ]
| clear H'1
| clear H'1 ]
| clear H'1 ]; auto.
exists c0; split; [ idtac | split; [ left | idtac ] ]; auto.
apply
(divp_trans _ _ _ _ _ _ _ _ _ cs n ltM)
with (y := ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM i c);
auto.
apply divP_ppc; auto.
apply divp_ppc; auto.
apply zerop_ddivp_ppc; auto.
rewrite H'5; auto.
exists c0; split; [ idtac | split; [ right | idtac ] ]; auto.
apply
(divp_trans _ _ _ _ _ _ _ _ _ cs n ltM)
with (y := ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM i c);
auto.
rewrite H'5; auto.
apply divP_ppc; auto.
apply divp_ppc; auto.
apply zerop_ddivp_ppc; auto.
rewrite <- H'5; auto.
apply divp_nzeropr with (1 := H'10); auto.
exists a; split; [ idtac | split; [ right | idtac ] ]; auto.
rewrite H'5; auto.
apply zerop_ddivp_ppc; auto.
elim (slice_inv i a L1 b); [ intros H'10 | intros H'10 | idtac ]; auto.
rewrite H' in H'10; simpl in H'10; auto.
lapply H'1;
[ intros H'3; elim (H'3 b);
[ intros c E; elim E; intros H'9 H'11; elim H'11; intros H'12 H'13;
elim H'12;
[ intros H'14; clear H'12 H'11 E H'1
| intros H'14; clear H'12 H'11 E H'1 ]
| clear H'1
| clear H'1 ]
| clear H'1 ]; auto.
exists c; split; [ right | idtac ]; auto.
exists c; split; [ idtac | split; [ right | idtac ] ]; auto.
elim (slice_cons i a L1 L2);
[ intros H'8; elim H'8; intros c E; elim E; intros H'9 H'11; clear E H'8
| intros H'8
| idtac ]; auto.
lapply H'1;
[ intros H'3; elim (H'3 c);
[ intros c0 E; elim E; intros H'12 H'13; elim H'13; intros H'14 H'15;
elim H'14;
[ intros H'16; clear H'14 H'13 E H'1
| intros H'16; clear H'14 H'13 E H'1 ]
| clear H'1
| clear H'1 ]
| clear H'1 ]; auto.
exists c0; split; [ idtac | split; [ left | idtac ] ]; auto.
apply
(divp_trans _ _ _ _ _ _ _ _ _ cs n ltM)
with (y := ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM i c);
auto.
apply
(divp_trans _ _ _ _ _ _ _ _ _ cs n ltM)
with (y := ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM i a);
auto.
apply divP_ppc; auto.
apply divp_ppc; auto.
apply zerop_ddivp_ppc; auto.
apply divP_ppc; auto.
apply divp_ppc; auto.
apply zerop_ddivp_ppc; auto.
apply divp_nzeropr with (1 := H'10); auto.
exists c0; split; [ idtac | split; [ right | idtac ] ]; auto.
apply
(divp_trans _ _ _ _ _ _ _ _ _ cs n ltM)
with (y := ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM i a);
auto.
apply divP_ppc; auto.
apply divp_ppc; auto.
apply zerop_ddivp_ppc; auto.
apply
(divp_trans _ _ _ _ _ _ _ _ _ cs n ltM)
with (y := ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM i c);
auto.
apply divP_ppc; auto.
apply divp_ppc; auto.
apply zerop_ddivp_ppc; auto.
apply divp_nzeropr with (1 := H'10); auto.
apply divp_nzeropr with (1 := H'11); auto.
exists a; split; [ idtac | split; [ right | idtac ] ]; auto.
Qed.

Lemma spolyp_cons_genPcP :
forall (aP R Q : list _) (i : poly A0 eqA ltM),
genPcP i aP R Q ->
~ BuchAux.zerop A A0 eqA n ltM i ->
forall b : poly A0 eqA ltM,
In b aP ->
~ BuchAux.zerop A A0 eqA n ltM b ->
exists c : poly A0 eqA ltM,
In c aP /\
(In
(spolyp A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM
ltM_dec os i c) Q \/
red A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
(spolyp A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM
ltM_dec os i c) (addEnd A A0 eqA n ltM i aP)) /\
divp A A0 eqA multA divA n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM i b) c.