Proof using.
induction xs; intros.
- auto with struct_util.
- invc_NoDup. simpl. break_if; eauto 6 using in_remove with struct_util.
Qed.

(* Specifications: *)
Lemma remove_length_ge : forall (x : A) xs,
NoDup xs ->
length (remove A_eq_dec x xs) >= length xs - 1.