intros i b work ds ni ai ai' a equiv_ins complete.
unfold deco_sound in |- *.
intros k a0 a1 b0 in_k.
elim (complete k a0 a1 b0 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_ai with i b ai; assumption.
Qed.

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