intros i work ds ni ai a a' equiv_ins complete.
unfold deco_sound in |- *.
intros k a0 a1 b in_k.
elim (complete k a0 a1 b 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_work_a with i a; assumption.
Qed.

Lemma deco_sound_shift_ni_x_ni_work :
forall (x : nested_imp) (work : nf_list) (ds : disjs)
(ni1 ni2 : nested_imps) (ai : atomic_imps) (a : atoms),
deco_sound work ds (ni1 ++ x :: ni2) ai a ->
deco_sound (NImp_NF (nested_imp2nimp x) :: work) ds (ni1 ++ ni2) ai a.