Proof using.
induction l; intros; simpl in *; intuition;
break_if; subst; simpl in *; intuition eauto.
Qed.

(* Specifications: *)
Lemma before_app :
forall x y l' l,
before (A := A) x y (l' ++ l) ->
~ In x l' ->
before (A := A) x y l.