Canonical polyOver_addrPred := AddrPred polyOver_addr_closed.
End PolyOverAdd.
Fact polyOverNr S (addS : zmodPred S) (kS : keyed_pred addS) :
oppr_closed (polyOver kS).
Canonical polyOver_opprPred S addS kS := OpprPred (@polyOverNr S addS kS).
Canonical polyOver_zmodPred S addS kS := ZmodPred (@polyOverNr S addS kS).
Section PolyOverSemiring.
Context (S : pred_class) (ringS : @semiringPred R 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}.