Ltac Esimpl :=
repeat (progress (
match goal with
| |- context [?P@?l] =>
match P with
| P0 => rewrite (Pphi0 l)
| P1 => rewrite (Pphi1 l)
| (mkPX ?P ?i ?n ?Q) => rewrite (mkPX_ok l P i n Q)
end
| |- context [[?c]] =>
match c with
| 0 => rewrite ring_morphism0
| 1 => rewrite ring_morphism1
| ?x + ?y => rewrite ring_morphism_add
| ?x * ?y => rewrite ring_morphism_mul
| ?x - ?y => rewrite ring_morphism_sub
| - ?x => rewrite ring_morphism_opp
end
end));
simpl; rsimpl.

Lemma PaddX_ok2 : forall P2,
(forall P l, (P2 ++ P) @ l == P2 @ l + P @ l)
/\
(forall P k n l,
(PaddX Padd P2 k n P) @ l ==
P2 @ l * pow_pos (nth 0 k l) n  + P @ l).