intros work i j ds ni ai a c in_ngamma0.
elim in_ngamma0; clear in_ngamma0 c.
intros n c0 nth; apply In_Work with n; assumption.
intros n i' j' nth; apply In_Disjs with (S n); apply My_NthS; 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_cons_ds_head :
forall (work : nf_list) (i j : Int) (ds : disjs) (ni : nested_imps)
(ai : atomic_imps) (a : atoms),
in_ngamma work ((i, j) :: ds) ni ai a (NDisj i j).