intros L1; elim L1; auto.
simpl in |- *; auto.
intros L2 p H; elim H.
simpl in |- *; unfold LetP in |- *; intros a l H' L2 p.
case
(zerop_dec A A0 eqA n ltM
(nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os
a (l ++ L2))).
intros H'0 H'1.
apply Cb_incl with (P := l ++ L2) (1 := cs); auto.
simpl in |- *; auto.
simpl in |- *.
intros H'0 H'1; case H'1;
[ intros H'2; rewrite <- H'2; clear H'1 | intros H'2; clear H'1 ];
auto.
apply Cb_cons; auto.
apply
Cb_trans_cons
with
(p := nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM
ltM_dec os a (l ++ L2)); auto.
apply
Cb_incl
with
(1 := cs)
(P := l ++
nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM
ltM_dec os a (l ++ L2) :: L2); auto.
change
(incl
(l ++
nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os
a (l ++ L2) :: L2)
(nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os
a (l ++ L2) :: a :: l ++ L2)) in |- *.
apply incl_app; auto with datatypes.
apply Cb_cons; auto.
Qed.
Definition Red (L : list (poly A0 eqA ltM)) : list (poly A0 eqA ltM) :=
redacc L nil.

Theorem Red_cb :
forall (L : list (poly A0 eqA ltM)) (p : poly A0 eqA ltM),
In p (Red L) -> Cb A A0 eqA plusA multA eqA_dec n ltM ltM_dec p L.