Fixpoint Ppow_pos (res P:Pol) (p:positive) : Pol :=
   match p with
   | xH => subst_l (res ** P)
   | xO p => Ppow_pos (Ppow_pos res P p) P p
   | xI p => subst_l ((Ppow_pos (Ppow_pos res P p) P p) ** P)
   end.Definition Ppow_N P n :=
   match n with
   | N0 => P1
   | Npos p => Ppow_pos P1 P p
   end.

(* Specifications: *)
Lemma Ppow_pos_ok l : (forall P, subst_l P@l == P@l) -> forall res P p, (Ppow_pos res P p)@l == res@l * (pow_pos Pmul P p)@l.
Lemma Ppow_N_ok l : (forall P, subst_l P@l == P@l) -> forall P n, (Ppow_N P n)@l == (pow_N P1 Pmul P n)@l.
Lemma Ppow_pos_ok l : (forall P, subst_l P@l == P@l) -> forall res P p, (Ppow_pos res P p)@l == res@l * (pow_pos Pmul P p)@l.
Lemma Ppow_N_ok l : (forall P, subst_l P@l == P@l) -> forall P n, (Ppow_N P n)@l == (pow_N P1 Pmul P n)@l.