intros x c x0 c0; generalize c c0; case x; case x0; simpl in |- *.
intros H' H'0; right; red in |- *; intros H'1; elim H'1.
intros a0 l H' H'0; right; red in |- *; intros H'1; elim H'1.
intros a0 l H' H'0; right; red in |- *; intros H'1; elim H'1.
intros a0 l a1 l0 H' H'0.
apply divP_dec with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l0); auto.
Qed.
Definition ppcp : poly A0 eqA ltM -> poly A0 eqA ltM -> poly A0 eqA ltM.
intros H'; case H'.
intros x; case x.
intros H'0 H'1; exists (pO A n); auto.
intros a l H'0 H'1; case H'1.
intros x0; case x0.
intros H'2; exists (pO A n); auto.
intros a0 l0 H'2; exists (ppc (A:=A) A1 (n:=n) a a0 :: pO A n).
change (canonical A0 eqA ltM (pX (ppc (A:=A) A1 (n:=n) a a0) (pO A n)))
in |- *; apply canonicalp1; auto.
apply ppc_nZ with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l0); auto.
Defined.

Theorem divp_ppc :
forall a b c : poly A0 eqA ltM, divp (ppcp a b) c -> divp (ppcp b a) c.