intros goal x work ds ni ai a context premiss.
unfold nsearch_spec in |- *; intros complete sound mon.
elim premiss; clear premiss.
intros ni1 less1 complete1 spec1.
apply NSearch_Res with (tail ni1).
inversion_clear less1; assumption.
generalize complete1; clear spec1 complete1.
inversion_clear less1; intros.
apply deco_sound_shift_ni_work with (x := Undecorated x); assumption.
apply deco_sound_shift_ni_work with (x := Decorated x k); assumption.
elim spec1; clear spec1.
intros derivable_goal; apply NDerivable; assumption.
intros k k_is_mon k_forces_ngamma k_nonforces_goal.
apply NRefutable with k; try assumption.
generalize k_forces_ngamma; clear complete1 k_forces_ngamma.
inversion_clear less1; intros.
apply forces_ngamma_shift_ni_work with (x := Undecorated x); assumption.
apply forces_ngamma_shift_ni_work with (x := Decorated x k0); assumption.
(* side premisses: Elim premiss.  *)
apply deco_sound_shift_work_ni0; assumption.
apply nsound_shift_work_ni; assumption.
apply nminimal_shift_work_ni; assumption.
Qed.

Lemma rule_shift_work_ai_new :
forall (goal i : Int) (b : normal_form) (work : nf_list)
(ds : disjs) (ni : nested_imps) (ai ai' : atomic_imps)
(a : atoms) (context : flist),
(forall x : unit, LOOKUP unit i a x -> False) ->
(forall bs : nf_list, LOOKUP nf_list i ai bs -> False) ->
EQUIV_INS nf_list i (cons b) nf_nil ai ai' ->
nsearch_spec goal work ds ni ai' a context ->
nsearch_spec goal (AImp i b :: work) ds ni ai a context.