Section before.
Variable A : Type.
Proof using.
induction l; intros; simpl in *; intuition.
Qed.
Proof using.
induction l; intros; simpl in *; intuition; subst; try congruence.
- exists nil. simpl. find_apply_lem_hyp in_split. break_exists. subst. eauto.
- exists nil. simpl. find_apply_lem_hyp in_split. break_exists. subst. eauto.
- exists nil. simpl. find_apply_lem_hyp in_split. break_exists. subst. eauto.
- eapply_prop_hyp In In; eauto. break_exists. subst.
exists (a :: x0), x1, x2. auto.
Qed.
Proof using.
induction xs; intros; simpl in *; intuition.
Qed.
Proof using.
induction xs; intros; simpl in *; intuition.
induction ys; intros; simpl in *; intuition.
Qed.
Proof using.
intros.
induction xs; intros; simpl in *; intuition.
Qed.
Proof using.
induction xs; intros; simpl in *; intuition; try congruence; eauto.
induction ys; intros; simpl in *; intuition; try congruence.
Qed.

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