Proof using.
induction xs; intros; simpl in *; intuition.
induction ys; intros; simpl in *; intuition.
Qed.

(* Specifications: *)
Lemma before_middle_insert :
forall xs y zs a b,
before(A:=A) a b (xs ++ zs) ->
b <> y ->
before a b (xs ++ y :: zs).