Proof with auto.
unfold compose, ext_eq.
unfold pick.
unfold lift.
simpl.
unfold ne_tree_monad.pick.
intros.
unfold compose.
simpl.
repeat rewrite ne_list.map_map...
Qed.

(* Specifications: *)
Lemma In_pick_inv (l: ne_list.L X) (r: prod m X):
ne_tree.In r (pick l) -> fst r = monoid_zero m /\ In (snd r) l.