intros goal i work ds ni ai a context in_atoms premiss.
unfold nsearch_spec in |- *; intros complete sound mon.
cut (EQUIV_INS unit i (fun _ => tt) tt a a).
intros equiv_ins.
elim premiss; clear premiss.
clear mon sound complete.
intros ni1 le1 complete1 spec1.
apply NSearch_Res with ni1; try assumption.
apply deco_sound_shift_a_work with a; assumption.
elim spec1; clear spec1.
intros der_goal; apply NDerivable; assumption.
intros k k_is_mon k_forces_ngamma k_notforces_goal.
apply NRefutable with k; try assumption.
apply forces_ngamma_shift_a_work with a; assumption.
(* side premisses: (Elim spec ..)  *)
apply deco_sound_shift_work_a with i a; assumption.
apply nsound_shift_work_a with i a; assumption.
apply nminimal_shift_work_a with i a; assumption.
(* side premiss (Cut (EQUIV_INS ....))  *)
clear complete sound mon premiss context ai ni ds work goal.
generalize in_atoms; clear in_atoms.
elim a; clear a; intros t avl_t.
unfold LOOKUP in |- *.
unfold EQUIV_INS in |- *.
intros lookup.
apply equiv_ins_intro.
intros k d eq_k lookup0.
rewrite (equal_eq k i eq_k); assumption.
intros k eq_k notlookup0.
rewrite (equal_eq k i eq_k); assumption.
intros; assumption.
intros; assumption.
Qed.
(**************************************************************************)

Lemma left_p_imp_work :
forall (goal i : Int) (b : normal_form) (work : nf_list)
(ds : disjs) (ni : nested_imps) (ai : atomic_imps)
(a : atoms) (context : flist),
forces0_t a i ->
nsearch_spec goal (b :: work) ds ni ai a context ->
nsearch_spec goal (AImp i b :: work) ds ni ai a context.