intros Q a b p q H'; inversion_clear H'; auto.
intros eq Cap.
elim (not_double_canonical A A0 eqA n ltM) with (a := b) (p := q); auto.
apply
eqp_imp_canonical
with
(p := pX b
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec a b0 nZb p q0))
(1 := cs); auto.
apply (canonical_pX_eqT A A0 eqA n ltM) with (a := a); auto.
apply canonical_spminusf_full_t with (1 := cs); auto.
apply inPolySet_imp_canonical with (L := Q); auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply nzeroP_comp_eqTerm with (1 := cs) (a := a); auto.
apply canonical_nzeroP with (p := p) (ltM := ltM); auto.
Qed.

Theorem reducetop_sp :
forall (Q : list (poly A0 eqA ltM)) (a b : Term A n)
(nZb : ~ zeroP (A:=A) A0 eqA (n:=n) b) (p q : list (Term A n)),
inPolySet (pX b q) Q ->
divP A A0 eqA multA divA n a b ->
reduce Q (pX a p)
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b nZb
p q).