intros work ds ni ai i a a' c equiv_ins 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 n; assumption.
intros n x' nth; apply In_Nested_Imps with n; assumption.
intros i0 b0 n bs lookup nth; apply In_Atomic_Imps with n bs; assumption.
intros i0 lookup.
apply In_Atoms.
generalize equiv_ins lookup; clear equiv_ins lookup.
elim a; clear a; intros t avl_t.
elim a'; clear a'; intros t' avl_t'.
simpl in |- *; intros equiv_ins lookup.
inversion_clear equiv_ins.
case (equal_dec i0 i).
intro equal.
apply H with tt; assumption.
intro notequal.
apply H1; assumption.
Qed.

Lemma in_ngamma_ins_a_head :
forall (work : nf_list) (ds : disjs) (ni : nested_imps)
(ai : atomic_imps) (i : Int) (a a' : atoms),
EQUIV_INS unit i (fun _ : unit => tt) tt a a' ->
in_ngamma work ds ni ai a' (NAtom i).