Proof using.
induction 1; intros.
- apply dup_drop_step_star_step_1. constructor. auto.
- concludes.
eapply dup_drop_step_star_trans; eauto.
apply dup_drop_cons.
apply dup_drop_step_star_step_1.
auto.
Qed.

(* Specifications: *)
Lemma dup_drop_step_star_remove_In :
forall l' l a,
In a l' ->
dup_drop_step_star l (remove A_eq_dec a l') ->
dup_drop_step_star (a :: l) l'.