Section POWER.
Variable subst_l : Pol -> Pol.
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.