intros i b work ds ni ai ai' a k notlookup equiv_ins k_forces_ngamma.
unfold forces_ngamma in |- *.
intros d in_gamma.
apply k_forces_ngamma.
apply in_ngamma_shift_work_ai_new with i b ai; assumption.
Qed.

Lemma forces_ngamma_shift_ai_work_old :
forall (i : Int) (b : normal_form) (work : nf_list)
(ds : disjs) (ni : nested_imps) (bs : nf_list) (ai ai' : atomic_imps)
(a : atoms) (k : kripke_tree),
LOOKUP nf_list i ai bs ->
EQUIV_INS nf_list i (cons b) nf_nil ai ai' ->
forces_ngamma work ds ni ai' a k ->
forces_ngamma (AImp i b :: work) ds ni ai a k.