Section PolyOverSemiring.
Variables (S : {pred R}) (ringS : semiringPred S) (kS : keyed_pred ringS).
Fact polyOver_mulr_closed : mulr_closed (polyOver kS).
Canonical polyOver_mulrPred := MulrPred polyOver_mulr_closed.
Canonical polyOver_semiringPred := SemiringPred polyOver_mulr_closed.

Lemma polyOverZ : {in kS & polyOver kS, forall c p, c *: p \is a polyOver kS}.