Proof with norm.
intros.
apply pred_ext.
apply andp_right. apply andp_left2... apply andp_left1...
apply andp_right. apply andp_left2... apply andp_left1...
Qed.

Lemma andp_derives {A} {NA: NatDed A}:
forall P Q P' Q': A, P |-- P' -> Q |-- Q' -> P && Q |-- P' && Q'.