Proof using.
induction 1.
- constructor.
- invc H.
+ eapply dup_drop_step_star_trans; [eauto|].
eapply dup_drop_step_star_step_1n; [eapply DDS_dup with (p := p); simpl; auto|].
auto using dup_drop_swap.
+ eapply dup_drop_step_star_trans; [eauto|].
eapply dup_drop_step_star_step_1n.
eapply DDS_drop with (xs := x :: xs) (p := p) (ys := ys).
constructor.
Qed.

Lemma dup_drop_Permutation :
forall l l',
Permutation l l' ->
dup_drop_step_star l l'.