intros i bs work ds ni ai ai' a k k_is_mon lookup equiv_del k_forces_ngamma.
apply forces_ngamma_del_ai_rev with i bs ai'; try assumption.
intros n b nth.
simpl in |- *; apply forces_b__forces_a_imp_b_t.
assumption.
apply k_forces_ngamma.
apply In_Work with n.
apply nth_app0; assumption.
apply forces_ngamma_app_work_tail with bs; assumption.
Qed.

(* Specifications: *)
Lemma forces_ngamma_shift_work_ai_strength :
forall (i : Int) (bs work : nf_list) (ds : disjs)
(ni : nested_imps) (ai ai' : atomic_imps) (a a' : atoms)
(k : kripke_tree),
Is_Monotone_kripke_tree k ->
EQUIV_INS unit i (fun _ : unit => tt) tt a a' ->
LOOKUP nf_list i ai bs ->
EQUIV_DEL nf_list i ai ai' ->
forces_ngamma work ds ni ai a' k ->
forces_ngamma (bs ++ work) ds ni ai' a' k.