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 grobner_def : forall L : list (poly A0 eqA ltM), Grobner A A0 A1 eqA plusA invA minusA multA divA eqA_dec n ltM ltM_dec L -> forall p : poly A0 eqA ltM, Cb A A0 eqA plusA multA eqA_dec n ltM ltM_dec p L -> zerop A A0 eqA n ltM p \/ (exists q : poly A0 eqA ltM, reducep L p q).