intros bs work ds ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth.
apply In_Work with (length bs + n); apply nth_app1; assumption.
intros n i j nth; apply In_Disjs with n; assumption.
intros n x nth; apply In_Nested_Imps with n; assumption.
intros i b n bs' lookup nth; apply In_Atomic_Imps with n bs'; assumption.
intros i lookup; apply In_Atoms; assumption.
Qed.

(* Specifications: *)
Lemma in_ngamma_work_app_rev :
forall (bs work : nf_list) (ds : disjs) (ni : nested_imps)
(ai : atomic_imps) (a : atoms) (c : normal_form),
in_ngamma (bs ++ work) ds ni ai a c ->
in_ngamma work ds ni ai a c + {n : nat | my_nth normal_form n bs c}.