intros L H'; inversion H'; auto.
intros p H'0.
case
(Reducef A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec
os L p).
intros x H'1; inversion H'.
lapply (H0 (s2p A A0 eqA n ltM p) (s2p A A0 eqA n ltM x));
[ intros H'4; lapply H'4; [ clear H'4 | clear H'4 ] | idtac ];
auto.
inversion H'1; auto.
inversion H1; auto.
intros H'2; left.
cut (eqP A eqA n (s2p A A0 eqA n ltM p) (pO A n)); auto.
case p; simpl in |- *; auto.
intros x1; case x1; auto.
intros a l H'3 H'4; inversion H'4.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (y := s2p A A0 eqA n ltM x);
auto.
intros H'2; right; cut (canonical A0 eqA ltM y); auto.
intros H'3; exists (mks A A0 eqA n ltM y H'3); generalize H5; case p;
simpl in |- *; auto.
apply canonical_reduce with (1 := cs) (3 := H5); auto.
generalize H'0; case p; simpl in |- *; auto.
generalize H'0; case p; simpl in |- *; auto.
Qed.

Theorem def_grobner :
forall L : list (poly A0 eqA ltM),
(forall p : poly A0 eqA ltM,
Cb A A0 eqA plusA multA eqA_dec n ltM ltM_dec p L ->
zerop A A0 eqA n ltM p \/ (exists q : poly A0 eqA ltM, reducep L p q)) ->
Grobner A A0 A1 eqA plusA invA minusA multA divA eqA_dec n ltM ltM_dec L.