Proof with auto.
induction l; simpl... intros [B C]. specialize (H a _ (conj B (IHl C))).
solve [rewrite listd_app; auto].
Qed.

(* Specifications: *)
Lemma listd_In_prop l : (forall a, In a l -> f a) -> listd f and True l.
Proof with simpl; try solve[exfalso; auto]; auto. induction l... Qed.
Lemma listd_In_inv_prop a l : listd f and True l -> In a l -> f a.
Proof with try solve[exfalso; auto]; auto.