(* Quicker than autorewrite with Esimpl :-) *)
 Ltac Esimpl := try rewrite_db Esimpl; rsimpl; simpl.

Lemma PaddX_ok P' P k l : (forall P l, (P++P')@l == P@l + P'@l) -> (PaddX Padd P' k P) @ l == P@l + P'@l * (hd l)^k.