(* File: niMinimal.v  (last edited on 27/10/2000) (c) Klaus Weich  *)
Require Export Forces_NGamma.
Require Export Derivable_Tools.
Definition nminimal (work : nf_list) (ds : disjs) (ni : nested_imps)
(ai : atomic_imps) (a : atoms) (context : flist) :=
forall (c : form) (k : kripke_tree),
Is_Monotone_kripke_tree k ->
forces_ngamma work ds ni ai a k -> In c context -> forces_t k c.
(*********************************************************************)
intros work ds ni ai a context k k_is_mon k_forces_ngamma mon c der_c.
elim der_c; clear der_c.
intros t der_t.
apply soundness_t with t context; try assumption.
intros c0 in_c0.
apply mon; assumption.
Qed.
(*********************************************************************)

Lemma nminimal_eqv :
forall (work : nf_list) (ds : disjs) (ni1 ni2 : nested_imps)
(ai : atomic_imps) (a : atoms) (context : flist),
eqv_ni ni1 ni2 ->
nminimal work ds ni1 ai a context -> nminimal work ds ni2 ai a context.