Variant rdvdp_spec p q : {poly R} -> bool -> Type :=
| Rdvdp k q1 & p * ((lead_coef q)^+ k)%:P = q1 * q : rdvdp_spec p q 0 true
| RdvdpN & rmodp p q != 0 : rdvdp_spec p q (rmodp p q) false.
(* Is that version useable ? *)

Lemma rdivp_eq d p :
(lead_coef d ^+ (rscalp p d)) *: p = (rdivp p d) * d + (rmodp p d).