Section PolyOverRing.
Variables (S : {pred R}) (ringS : subringPred S) (kS : keyed_pred ringS).
Canonical polyOver_smulrPred := SmulrPred (polyOver_mulr_closed kS).
Canonical polyOver_subringPred := SubringPred (polyOver_mulr_closed kS).

Lemma polyOver_deriv S (ringS : semiringPred S) (kS : keyed_pred ringS) :
{in polyOver kS, forall p, p^`() \is a polyOver kS}.