intros L H'.
apply Grobner0.
intros p q H'0 H'1.
cut (canonical A0 eqA ltM p).
intros H'2.
cut (canonical A0 eqA ltM q).
intros H'3.
elim (H' (mks A A0 eqA n ltM q H'3)); [ intros H'6 | intros H'6 | idtac ];
auto.
generalize H'3 H'6; case q; simpl in |- *; auto.
intros a l H'4 H'5; elim H'5; auto.
case H'6; intros q0 E; clear H'6.
inversion H'1.
generalize H0 E; case q0; auto.
simpl in |- *; auto.
intros x H'4 H'5 H'6;
absurd
(reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec L q x);
auto.
simpl in |- *; auto.
apply reducestar_cb with (a := p) (1 := cs); auto.
inversion H'1.
apply canonical_reduceplus with (1 := cs) (3 := H); auto.
apply CombLinear_canonical with (1 := cs) (3 := H'0); auto.
Qed.

Theorem reduce_divp :
forall (p q : poly A0 eqA ltM) (Q : list (poly A0 eqA ltM)),
reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q
(s2p A A0 eqA n ltM p) (s2p A A0 eqA n ltM q) ->
exists r : poly A0 eqA ltM,
In r (q :: Q) /\
divp A A0 eqA multA divA n ltM p r /\ ~ zerop A A0 eqA n ltM r.