intros Q p H' H'0.
elim
reduce0_reducestar
with
(1 := cs)
(eqA_dec := eqA_dec)
(ltM_dec := ltM_dec)
(Q := Q)
(p := pO A n); auto.
intros t E; apply reducestar0; auto.
apply pO_irreducible; auto.
Qed.
Definition spO : poly A0 eqA ltM.
exists (pO A n); simpl in |- *; auto.
Defined.
Definition sp1 : poly A0 eqA ltM.
exists (pX (A1, M1 n) nil); auto.
Defined.
Definition sgen : nat -> poly A0 eqA ltM.
intros m; exists (pX (A1, gen_mon n m) (pO A n)).
apply canonicalp1; auto.
Defined.
Definition sscal : A -> poly A0 eqA ltM -> poly A0 eqA ltM.
intros a p; case p.
intros x H'1; exists (tmults A0 multA eqA_dec (a, M1 n) x); auto.
unfold tmults in |- *; case (zeroP_dec A A0 eqA eqA_dec n (a, M1 n));
simpl in |- *; auto.
Qed.

Theorem red_cons :
forall (a : poly A0 eqA ltM) (p : list (poly A0 eqA ltM)), In a p -> red a p.