Proof using.
induction ys; intros; simpl in *; intuition.
break_if; subst; simpl in *; intuition.
specialize (IHys a0 (remove A_eq_dec a l)). intuition.
Qed.

(* Specifications: *)
Lemma before_remove_all :
forall x y l ys,
before x y (remove_all ys l) ->
~ In y ys ->
before x y l.