intros i bs work ds ni ai ai' a 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_weak with i bs ai'; assumption.
Qed.

Lemma deco_sound_shift_work_ai_strength :
forall (i : Int) (bs work : nf_list) (ds : disjs)
(ni : nested_imps) (ai ai' : atomic_imps) (a a' : atoms),
EQUIV_INS unit i (fun _ : unit => tt) tt a a' ->
LOOKUP nf_list i ai bs ->
EQUIV_DEL nf_list i ai ai' ->
deco_sound work ds ni ai a' -> deco_sound (bs ++ work) ds ni ai' a'.