intros x work ds ni ai a complete.
unfold deco_sound in |- *.
intros k i0 i1 j in_k.
elim (complete k i0 i1 j); clear complete.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_shift_work_ni; assumption.
inversion_clear in_k.
discriminate H.
assumption.
Qed.

Lemma deco_sound_shift_ai_work_new :
forall (i : Int) (b : normal_form) (work : nf_list)
(ds : disjs) (ni : nested_imps) (ai ai' : atomic_imps)
(a : atoms),
(forall bs : nf_list, LOOKUP nf_list i ai bs -> False) ->
EQUIV_INS nf_list i (cons b) nf_nil ai ai' ->
deco_sound work ds ni ai' a -> deco_sound (AImp i b :: work) ds ni ai a.