Proof using.
induction l; intros; simpl in *; intuition.
- rewrite remove_all_nil in *. simpl in *. intuition.
- pose proof remove_all_cons ys a l. intuition.
+ repeat find_rewrite. right. intuition eauto.
subst; intuition.
+ repeat find_rewrite. simpl in *. intuition eauto.
Qed.

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