intros work ds ni i b ai ai' a c notlookup equiv_ins in_ngamma0.
elim (in_ngamma_cons_work_rev (AImp i b) work ds ni ai a c in_ngamma0);
clear in_ngamma0.
intros in_ngamma0.
apply in_ngamma_ins_ai_tail with i b ai; assumption.
intros eq_c;  rewrite eq_c.
apply in_ngamma_ins_ai_head_new with ai; assumption.
Qed.

(* Specifications: *)
Lemma in_ngamma_shift_work_ai_old :
forall (work : nf_list) (ds : disjs) (ni : nested_imps)
(i : Int) (b : normal_form) (bs : nf_list) (ai ai' : atomic_imps)
(a : atoms) (c : normal_form),
LOOKUP nf_list i ai bs ->
EQUIV_INS nf_list i (cons b) nf_nil ai ai' ->
in_ngamma (AImp i b :: work) ds ni ai a c -> in_ngamma work ds ni ai' a c.