Proof using.
induction 1.
- constructor.
- auto using dup_drop_cons.
- auto using dup_drop_swap.
- eauto using dup_drop_step_star_trans.
Qed.

(* Specifications: *)
Lemma remove_not_in_nop :
forall a l,
~ In a l ->
remove A_eq_dec a l = l.