Fixpoint mkmult_pow (r x:R) (p: positive) {struct p} : R :=
match p with
| xH => r*x
| xO p => mkmult_pow (mkmult_pow r x p) x p
| xI p => mkmult_pow (mkmult_pow (r*x) x p) x p
end.
Definition mkpow x p :=
match p with
| xH => x
| xO p => mkmult_pow x x (Pos.pred_double p)
| xI p => mkmult_pow x x (xO p)
end.
Definition mkopp_pow x p :=
match p with
| xH => -x
| xO p => mkmult_pow (-x) x (Pos.pred_double p)
| xI p => mkmult_pow (-x) x (xO p)
end.
Definition Pphi_dev := Pphi_avoid mkpow mkopp_pow mkmult_pow.

(* Specifications: *)
Lemma mkmult_pow_ok p r x : mkmult_pow r x p == r * x^p.
Lemma mkpow_ok p x : mkpow x p == x^p.
Lemma mkopp_pow_ok p x : mkopp_pow x p == - x^p.
Lemma Pphi_dev_ok : forall P fv, Pphi_dev fv P == P@fv.
Lemma ring_rw_correct : forall n lH l,
interp_PElist l lH ->
forall lmp, mk_monpol_list lH = lmp ->
forall pe npe, norm_subst n lmp pe = npe ->
PEeval l pe == Pphi_dev l npe.