intros b c work ds ni ai a forces_bc complete.
unfold deco_sound in |- *.
intros k i0 i1 b0 in_k.
elim (complete k i0 i1 b0 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_weak with b; try assumption.
intros forces_b.
apply forces_bc; assumption.
Qed.

Lemma deco_sound_shift_work_ai_weak :
forall (i : Int) (bs work : nf_list) (ds : disjs)
(ni : nested_imps) (ai ai' : atomic_imps) (a : atoms),
LOOKUP nf_list i ai bs ->
EQUIV_DEL nf_list i ai ai' ->
deco_sound (bs ++ work) ds ni ai' a -> deco_sound work ds ni ai a.