Proof using.
induction xs; intros.
- auto.
- simpl in *.
specialize (IHxs eq_dec).
break_if; subst; simpl; omega.
Qed.

(* Specifications: *)
Lemma remove_length_lt :
forall (x : A) xs eq_dec,
In x xs ->
length xs > length (remove eq_dec x xs).