intros p L1 L2 H' q; case p; case q; simpl in |- *; auto.
intros x H'0 x0 c H'1; generalize c; unfold reducep in H'1; simpl in H'1;
elim H'1; auto.
intros a b nZb p0 q0 r H'2 H'3 H'4 c0.
case inPolySet_inv1 with (1 := H'2).
intros q1 E; case E; intros H4 H5; clear E.
elim (H' q1);
[ intros r2 E; elim E; intros H'10 H'11; clear E | idtac | idtac ];
auto.
generalize H'10 H'11 H5 c0 H4; case q1; simpl in |- *; auto.
intros x1 c1; case r2; simpl in |- *; auto.
intros x2; case x2; simpl in |- *; auto.
generalize c1; case x1; auto.
intros c2 c3 H'6 H'8 H'9 c4 H'12; discriminate.
intros t l c2 c3 H'6 H'8; elim H'8.
intros a0 l c2; generalize c1; case x1; auto.
intros c3 H'6 H'8; elim H'8.
intros t l0 c3 H'6 H'8 H'9 c4 H'12.
cut (~ zeroP (A:=A) A0 eqA (n:=n) a0); [ intros nZa0 | idtac ].
cut
(canonical A0 eqA ltM
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a a0
nZa0 p0 l)); [ intros Z | idtac ].
exists
(mks A A0 eqA n ltM
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a a0
nZa0 p0 l) Z); simpl in |- *; auto.
red in |- *; simpl in |- *; apply reducetop_sp with (1 := cs); auto.
change
(inPolySet A A0 eqA n ltM
(s2p A A0 eqA n ltM (mks A A0 eqA n ltM (pX a0 l) c2)) L2)
in |- *.
apply in_inPolySet; auto.
simpl in |- *.
red in |- *; intros H'13; inversion H'13.
injection H'9; intros.
apply (divP_trans _ _ _ _ _ _ _ _ _ cs n) with (y := b); auto.
rewrite H0; auto.
apply canonical_spminusf_full with (1 := cs); auto.
injection H'9; intros.
apply (divP_trans _ _ _ _ _ _ _ _ _ cs n) with (y := b); auto.
rewrite H0; auto.
apply canonical_nzeroP with (ltM := ltM) (p := l); auto.
generalize H5; case q1; simpl in |- *; auto.
intros x1; case x1; simpl in |- *; auto.
intros c1 H'6; discriminate.
intros a b p0 q0 H'2 H'3 H'4 c0.
cut (canonical A0 eqA ltM p0);
[ intro | apply canonical_imp_canonical with (a := a); auto ].
case (H'3 H); intros r; case r.
intros x1 H'6 H'7.
cut (canonical A0 eqA ltM (pX b x1)); [ intros Z | idtac ].
exists (mks A A0 eqA n ltM (pX b x1) Z); simpl in |- *; auto.
red in |- *; simpl in |- *; red in H'7; simpl in H'7; auto.
apply eqp_imp_canonical with (1 := cs) (p := pX a x1); auto.
apply ltP_pX_canonical; auto.
apply canonical_nzeroP with (ltM := ltM) (p := p0); auto.
apply ltP_trans with (y := p0); auto.
apply ltP_reduce with (1 := cs) (3 := H'7); auto.
apply (canonical_pX_ltP A A0 eqA); auto.
Qed.

Theorem nf_divp_zero :
forall (p : poly A0 eqA ltM) (L : list (poly A0 eqA ltM)),
~ zerop A A0 eqA n ltM p ->
zerop A A0 eqA n ltM
(nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os p
L) ->
exists q : poly A0 eqA ltM,
In q L /\ divp A A0 eqA multA divA n ltM p q /\ ~ zerop A A0 eqA n ltM q.