Definition reducep (L : list (poly A0 eqA ltM)) (p q : poly A0 eqA ltM) :
  Prop :=
  reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec L
    (s2p A A0 eqA n ltM p) (s2p A A0 eqA n ltM q).

Theorem reduce_divp : forall (p q : poly A0 eqA ltM) (Q : list (poly A0 eqA ltM)), reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q (s2p A A0 eqA n ltM p) (s2p A A0 eqA n ltM q) -> exists r : poly A0 eqA ltM, In r (q :: Q) /\ divp A A0 eqA multA divA n ltM p r /\ ~ zerop A A0 eqA n ltM r.