intros c work ds ni ai a 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_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_cons_work_tail with c; assumption.
Qed.

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