intros b c d work ds ni ai a k k_is_mon forces_bcd k_forces_ngamma.
apply forces_ngamma_cons_work.
apply forces_bcd.
apply k_forces_ngamma.
apply in_ngamma_cons_work_head.
apply k_forces_ngamma.
apply in_ngamma_cons_work_tail.
apply in_ngamma_cons_work_head.
apply forces_ngamma_cons_work_tail with c.
apply forces_ngamma_cons_work_tail with b; assumption.
Qed.

Lemma forces_ngamma_shift_work_ai_weak :
forall (i : Int) (bs work : nf_list) (ds : disjs)
(ni : nested_imps) (ai ai' : atomic_imps) (a : atoms)
(k : kripke_tree),
Is_Monotone_kripke_tree k ->
LOOKUP nf_list i ai bs ->
EQUIV_DEL nf_list i ai ai' ->
forces_ngamma (bs ++ work) ds ni ai' a k -> forces_ngamma work ds ni ai a k.