intros Q p q H' H'0 H'1.
elim
(red_plus_zero_reduce Q p
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) q));
[ intros r1 E; elim E; intros H'8 H'9; clear E | idtac | idtac | idtac ];
auto.
exists r1; split; auto.
apply
reduceplus_eqp_com
with
(p := q)
(q := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
r1)); auto.
apply reduceplus_mults_invr; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n)
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n))
(invTerm (A:=A) invA (n:=n) (T1 A1 n))) r1);
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply
reduceplus_eqp_com
with
(p := minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p q)
(q := pO A n); auto.
Qed.

(* Specifications: *)
Theorem reduce_plus_top :
forall (Q : list (poly A0 eqA ltM)) (r : list (Term A n)),
canonical A0 eqA ltM r ->
forall (a b : Term A n) (nZb : ~ zeroP (A:=A) A0 eqA (n:=n) b)
(p q : list (Term A n)),
inPolySet A A0 eqA n ltM (pX b q) Q ->
canonical A0 eqA ltM (pX a p) ->
divP A A0 eqA multA divA n a b ->
ex
(fun s : list (Term A n) =>
reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p) r) s /\
reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p q) r) s).