intros x x' ni1 ni2 in_ni_x_ni.
elim (in_app_or_ni x ni1 (x' :: ni2) in_ni_x_ni); clear in_ni_x_ni.
intros in_ni1; left; apply in_ni0_in_nini; assumption.
intros in_ni2; inversion_clear in_ni2.
right.
symmetry  in |- *; assumption.
left; apply in_ni1_in_nini; assumption.
Qed.

Lemma in_ni_x_ni_tail :
forall (x x' : nested_imp) (ni1 ni2 : nested_imps),
In x (ni1 ++ ni2) -> In x (ni1 ++ x' :: ni2).