Section PolyOverRing.
Context (S : pred_class) (ringS : @subringPred R S) (kS : keyed_pred ringS).
Canonical polyOver_smulrPred := SmulrPred (polyOver_mulr_closed kS).
Canonical polyOver_subringPred := SubringPred (polyOver_mulr_closed kS).

(* Specifications: *)
Lemma polyOverXsubC c : ('X - c%:P \in polyOver kS) = (c \in kS).
Lemma polyOver_deriv S (ringS : semiringPred S) (kS : keyed_pred ringS) :
{in polyOver kS, forall p, p^`() \is a polyOver kS}.
Lemma derivC c : c%:P^`() = 0.
Fact deriv_is_linear : linear deriv.