Proof using.
intros a l1 l2; generalize a l1; elim l2; clear a l1 l2; simpl in |- *; auto.
intros a l1 H1; case H1.
intros a l H a0 l1 [H0| H0].
injection H0; intros H1 H2; rewrite H2; rewrite H1; auto.
generalize H H0; elim (split_one l); simpl in |- *; auto.
intros H1 H2; case H2.
intros a1 l0 H1 H2 [H3| H3]; auto.
injection H3; intros H4 H5; (rewrite <- H4; rewrite <- H5).
apply permutation_trans with (a :: fst a1 :: snd a1); auto.
apply permutation_skip.
apply H2; auto.
case a1; simpl in |- *; auto.
Qed.

(* All elements of the list are there *)
Theorem split_one_in_ex :
forall (a : A) (l1 : list A),
In a l1 -> exists l2 : list A, In (a, l2) (split_one l1).