Hint Rewrite
Pphi0
Pphi1
mkPinj_ok
mkPX_ok
(morph0 CRmorph)
(morph1 CRmorph)
(morph0 CRmorph)
(morph_add CRmorph)
(morph_mul CRmorph)
(morph_sub CRmorph)
(morph_opp CRmorph)
: Esimpl.
(* Quicker than autorewrite with Esimpl :-) *)
Ltac Esimpl := try rewrite_db Esimpl; rsimpl; simpl.

Lemma PsubX_ok P' P k l :
(forall P l, (P--P')@l == P@l - P'@l) ->
(PsubX Psub P' k P) @ l == P@l - P'@l * (hd l)^k.