Remark nsound_del_ai :
 forall (i : Int) (work : nf_list) (ds : disjs) (ni : nested_imps)
   (ai ai' : atomic_imps) (a : atoms) (context : flist),
 EQUIV_DEL nf_list i ai ai' ->
 nsound work ds ni ai a context -> nsound work ds ni ai' a context.

(* Specifications: *)
Lemma nsound_cons_work_cons_context : forall (c : normal_form) (work : nf_list) (ds : disjs) (ni : nested_imps) (ai : atomic_imps) (a : atoms) (context : flist), nsound work ds ni ai a context -> nsound (c :: work) ds ni ai a (nf2form c :: context).
Lemma nsound_cons_work_weak : forall (b c : normal_form) (work : nf_list) (ds : disjs) (ni : nested_imps) (ai : atomic_imps) (a : atoms) (context : flist), (Derivable context (nf2form b) -> Derivable context (nf2form c)) -> nsound (b :: work) ds ni ai a context -> nsound (c :: work) ds ni ai a context.
Lemma nsound_shift_work_ai_strength : forall (i : Int) (bs work : nf_list) (ds : disjs) (ni : nested_imps) (ai ai' : atomic_imps) (a a' : atoms) (context : flist), EQUIV_INS unit i (fun _ : unit => tt) tt a a' -> LOOKUP nf_list i ai bs -> EQUIV_DEL nf_list i ai ai' -> nsound work ds ni ai a' context -> nsound (bs ++ work) ds ni ai' a' context.