intros i b work ds ni ai ai' a notlookup equiv_ins complete.
unfold deco_sound in |- *.
intros k a0 a1 b0 in_k.
elim (complete k a0 a1 b0 in_k); clear complete.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_shift_ai_work_new with ai'; assumption.
Qed.

(* Specifications: *)
Lemma deco_sound_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),
LOOKUP nf_list i ai bs ->
EQUIV_INS nf_list i (cons b) nf_nil ai ai' ->
deco_sound work ds ni ai' a -> deco_sound (AImp i b :: work) ds ni ai a.