Inductive reducehead (Q : list (poly A0 eqA ltM)) :
list (Term A n) -> list (Term A n) -> Prop :=
    reduceheadO :
      forall (a b : Term A n) (nZb : ~ zeroP (A:=A) A0 eqA (n:=n) b)
        (p q : list (Term A n)),
      pickinSetp a (pX b q) Q ->
      reducehead Q (pX a p)
        (spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
           b nZb p q).Inductive reducehead (Q : list (poly A0 eqA ltM)) :
list (Term A n) -> list (Term A n) -> Prop :=
    reduceheadO :
      forall (a b : Term A n) (nZb : ~ zeroP (A:=A) A0 eqA (n:=n) b)
        (p q : list (Term A n)),
      pickinSetp a (pX b q) Q ->
      reducehead Q (pX a p)
        (spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
           b nZb p q).

Theorem reducehead_imp_reduce : forall (Q : list (poly A0 eqA ltM)) (p q : list (Term A n)), reducehead Q p q -> reduce Q p q.