intros i j work ds ni ai a 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_ds_work; assumption.
Qed.

Lemma deco_sound_shift_work_ds :
forall (i j : Int) (work : nf_list) (ds : disjs) (ni : nested_imps)
(ai : atomic_imps) (a : atoms),
deco_sound (NDisj i j :: work) ds ni ai a ->
deco_sound work ((i, j) :: ds) ni ai a.