intros i bs work ds ni ai ai' a a' equiv_ins lookup equiv_del complete.
unfold deco_sound in |- *.
intros k i0 i1 j in_k.
elim (complete k i0 i1 j in_k); clear complete in_k.
intros k_is_mon k_forces_ngamma k_nonforces.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_shift_work_ai_strength with i ai a; assumption.
Qed.

Lemma deco_sound_inf :
forall (ni ni1 ni2 : nested_imps) (work : nf_list)
(ds : disjs) (ai : atomic_imps) (a : atoms),
le_ni ni ni1 ->
eqv_ni ni ni2 ->
(forall (x : nimp) (k : kripke_tree),
In (Decorated x k) ni -> In (Decorated x k) ni1 \/ In (Decorated x k) ni2) ->
deco_sound work ds ni1 ai a ->
deco_sound work ds ni2 ai a -> deco_sound work ds ni ai a.