intros goal i b work ds ni bs ai ai' a context notlookup_i lookup_bs
equiv_ins 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_ai_work_old with bs ai'; 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_ai_work_old with bs ai'; assumption.
(* side premisses: Elim premiss.  *)
apply deco_sound_shift_work_ai with i b ai; assumption.
apply nsound_shift_work_ai with i b ai; assumption.
apply nminimal_shift_work_ai_old with i b bs ai; assumption.
Qed.

Lemma rule_shift_work_a :
forall (goal i : Int) (work : nf_list) (ds : disjs)
(ni : nested_imps) (ai : atomic_imps) (a a' : atoms)
(context : flist),
(forall bs : nf_list, LOOKUP nf_list i ai bs -> False) ->
(forall d : unit, LOOKUP unit i a d -> False) ->
EQUIV_INS unit i (fun _ : unit => tt) tt a a' ->
nsearch_spec goal work ds ni ai a' context ->
nsearch_spec goal (NAtom i :: work) ds ni ai a context.