(* 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.
(*********************************************************************)
intros work ds ni1 ni2 ai a context eq12 mon.
unfold nminimal in |- *.
intros c k k_is_mon k_forces in_a.
apply mon; try assumption.
apply forces_ngamma_eqv with ni2; assumption.
Qed.
(************************************************************************)
intros i j work ds ni ai a context mon.
unfold nminimal in |- *.
intros c k k_is_mon k_forces_ngamma in_a.
apply mon; try assumption.
apply forces_ngamma_shift_ds_work; assumption.
Qed.
intros x work ds ni ai a context mon.
unfold nminimal in |- *.
intros c k k_is_mon k_forces_ngamma in_a.
apply mon; try assumption.
apply forces_ngamma_shift_ni_work; assumption.
Qed.
intros i b work ds ni ai ai' a context lookup equiv_ins mon.
unfold nminimal in |- *.
intros c k k_is_mon k_forces_ngamma in_a.
apply mon; try assumption.
apply forces_ngamma_shift_ai_work_new with ai'; assumption.
Qed.
intros i b work ds ni bs ai ai' a context notlookup equiv_ins mon.
unfold nminimal in |- *.
intros c k k_is_mon k_forces_ngamma in_a.
apply mon; try assumption.
apply forces_ngamma_shift_ai_work_old with bs ai'; assumption.
Qed.
intros i work ds ni ai a a' context equiv_ins mon.
unfold nminimal in |- *.
intros c k k_is_mon k_forces_ngamma in_a.
apply mon; try assumption.
apply forces_ngamma_shift_a_work with a'; assumption.
Qed.
intros x work ds ni1 ni2 ai a context min.
unfold nminimal in |- *.
intros c k k_is_mon k_forces_ngamma in_c.
apply min; try assumption.
apply forces_ngamma_shift_ni_x_ni_work; assumption.
Qed.
intros x work ds ni1 ni2 ai a context min.
unfold nminimal in |- *.
intros c k k_is_mon k_forces_ngamma in_c.
apply min; try assumption.
apply forces_ngamma_shift_work_ni_x_ni; assumption.
Qed.
(********************************************************************)
intros b c work ds ni ai a context forces_cb minimal.
unfold nminimal in |- *.
intros c0 k k_is_mon k_forces_ngamma in_c0.
apply minimal; try assumption.
apply forces_ngamma_cons_work_weak with c; try assumption.
intros forces_c.
apply forces_cb; assumption.
Qed.
intros b c work ds ni ai a context forces_cb minimal.
unfold nminimal in |- *.
intros c0 k k_is_mon k_forces_ngamma in_c0.
apply minimal; try assumption.
apply forces_ngamma_cons_work_weak with c; try assumption.
intros forces_c.
apply forces_cb; assumption.
Qed.

Lemma nminimal_shift_work_ai_weak :
forall (i : Int) (bs work : nf_list) (ds : disjs)
(ni : nested_imps) (ai ai' : atomic_imps) (a : atoms)
(context : flist),
LOOKUP nf_list i ai bs ->
EQUIV_DEL nf_list i ai ai' ->
nminimal work ds ni ai a context ->
nminimal (bs ++ work) ds ni ai' a context.