intros goal i j 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 ni1.
assumption.
apply deco_sound_shift_ds_work; 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.
apply forces_ngamma_shift_ds_work; assumption.
(* side premisses: Elim premiss. *)
apply deco_sound_shift_work_ds; assumption.
apply nsound_shift_work_ds; assumption.
apply nminimal_shift_work_ds; assumption.
Qed.

(* Specifications: *)
Lemma rule_shift_work_ni0 :
forall (goal : Int) (x : nimp) (work : nf_list) (ds : disjs)
(ni : nested_imps) (ai : atomic_imps) (a : atoms)
(context : flist),
nsearch_spec goal work ds (Undecorated x :: ni) ai a context ->
nsearch_spec goal (NImp_NF x :: work) ds ni ai a context.