intros a b P; elim P; simpl in |- *; auto.
unfold slicef in |- *; simpl in |- *; auto.
case (foreigner_dec A A0 A1 eqA multA n ltM a b); intros H; apply incl_refl;
auto with datatypes.
intros c; unfold slicef in |- *; simpl in |- *.
case
(divp_dec _ _ _ _ _ _ _ _ _ cs n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM a b) c);
auto with datatypes.
case
(divp_dec _ _ _ _ _ _ _ _ _ cs n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM a c) b);
auto with datatypes.
intros H' H'0 l; case (slice a b l); simpl in |- *; auto with datatypes.
Qed.

Theorem slice_inv :
forall (a b : poly A0 eqA ltM) (P : list (poly A0 eqA ltM))
(c : poly A0 eqA ltM),
In c P ->
In c (getRes (slice a b P)) \/
divp A A0 eqA multA divA n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM a c) b.