intros work i j ds ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth; left; apply In_Work with n; assumption.
intros n; case n; clear n.
intros i' j' nth; right; inversion_clear nth; trivial.
intros n i' j' nth; left; apply In_Disjs with n; inversion_clear nth;
assumption.
intros n x nth; left; apply In_Nested_Imps with n; 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.

Lemma in_ngamma_cons_ni_tail :
forall (work : nf_list) (ds : disjs) (x : nested_imp)
(ni : nested_imps) (ai : atomic_imps) (a : atoms)
(c : normal_form),
in_ngamma work ds ni ai a c -> in_ngamma work ds (x :: ni) ai a c.