Proof using.
induction xs; intros.
- auto.
- invc_NoDup. simpl. break_if.
+ rewrite <- minus_n_O.
subst.
rewrite remove_length_not_in; auto.
+ simpl. concludes. omega.
Qed.

Lemma remove_length_le :
forall (x : A) xs eq_dec,
length xs >= length (remove eq_dec x xs).