Inductive dup_drop_step : list A -> list A -> Prop :=
  | DDS_dup : forall l p,
      In p l ->
      dup_drop_step l (p :: l)
  | DDS_drop : forall xs p ys,
      dup_drop_step (xs ++ p :: ys) (xs ++ ys).Inductive dup_drop_step : list A -> list A -> Prop :=
  | DDS_dup : forall l p,
      In p l ->
      dup_drop_step l (p :: l)
  | DDS_drop : forall xs p ys,
      dup_drop_step (xs ++ p :: ys) (xs ++ ys).Inductive dup_drop_step : list A -> list A -> Prop :=
  | DDS_dup : forall l p,
      In p l ->
      dup_drop_step l (p :: l)
  | DDS_drop : forall xs p ys,
      dup_drop_step (xs ++ p :: ys) (xs ++ ys).Definition dup_drop_step_star := clos_refl_trans_n1 _ dup_drop_step.

Lemma remove_In_elim : forall x a l, In x (remove A_eq_dec a l) -> x <> a /\ In x l.