intros work ds x ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c1 nth; left; apply In_Work with n; assumption.
intros n i' j' nth; left; apply In_Disjs with n; assumption.
intros n; case n; clear n.
intros x' nth; right; inversion_clear nth; trivial.
intros n x' nth; left; apply In_Nested_Imps with n; inversion_clear nth;
assumption.
intros i' b n bs lookup nth; left; apply In_Atomic_Imps with n bs; assumption.
intros i' lookup; left; apply In_Atoms; assumption.
Qed.

(* Specifications: *)
Lemma in_ngamma_ni_x_ni_head :
forall (work : nf_list) (ds : disjs) (x : nested_imp)
(ni1 ni2 : nested_imps) (ai : atomic_imps) (a : atoms),
in_ngamma work ds (ni1 ++ x :: ni2) ai a (NImp_NF (nested_imp2nimp x)).