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(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
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(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(* définition inductive de la coaccessibilité *)
Inductive coacc : preDTA -> ad -> ad -> Prop :=
| coacc_id :
forall (d : preDTA) (a : ad) (s : state),
MapGet state d a = Some s -> coacc d a a
| coacc_nxt :
forall (d : preDTA) (a0 a1 a2 : ad) (s1 s2 : state)
(pl : prec_list) (c : ad),
MapGet state d a2 = Some s2 ->
MapGet state d a1 = Some s1 ->
MapGet prec_list s1 c = Some pl ->
prec_occur pl a2 -> coacc d a0 a1 -> coacc d a0 a2.
Definition coacc_transitive_def (d : preDTA) (a0 a1 : ad) : Prop :=
forall a2 : ad, coacc d a0 a1 -> coacc d a2 a0 -> coacc d a2 a1.
(* fonction dont on va calculer le point fixe : *)
Fixpoint map_replace (A : Set) (m : Map A) {struct m} :
ad -> A -> Map A :=
fun (a : ad) (x : A) =>
match m with
| M0 => M0 A
| M1 b y => if N.eqb a b then M1 A b x else M1 A b y
| M2 m n =>
match a with
| N0 => M2 A (map_replace A m N0 x) n
| Npos q =>
match q with
| xH => M2 A m (map_replace A n N0 x)
| xO p => M2 A (map_replace A m (Npos p) x) n
| xI p => M2 A m (map_replace A n (Npos p) x)
end
end
end.
Fixpoint map_or (m0 m1 : Map bool) {struct m1} : Map bool :=
match m0, m1 with
| M0, _ => M0 bool
| _, M0 => M0 bool
| M1 a0 b0, M1 a1 b1 =>
if N.eqb a0 a1 then M1 bool a0 (b0 || b1) else M0 bool
| M1 _ _, M2 _ _ => M0 bool
| M2 _ _, M1 _ _ => M0 bool
| M2 x0 y0, M2 x1 y1 => M2 bool (map_or x0 x1) (map_or y0 y1)
end.
Fixpoint pl_coacc (d : preDTA) (pl : prec_list) {struct pl} :
Map bool :=
match pl with
| prec_empty => map_mini state d
| prec_cons a la ls =>
map_replace bool (map_or (pl_coacc d la) (pl_coacc d ls)) a true
end.
Fixpoint st_coacc (d : preDTA) (s : state) {struct s} :
Map bool :=
match s with
| M0 => map_mini state d
| M1 a pl => pl_coacc d pl
| M2 x y => map_or (st_coacc d x) (st_coacc d y)
end.
Fixpoint predta_coacc_0 (d d' : preDTA) {struct d'} :
Map bool -> Map bool :=
fun m : Map bool =>
match d', m with
| M0, M0 => map_mini state d
| M1 a s, M1 a' b =>
if N.eqb a a' && b then st_coacc d s else map_mini state d
| M2 x y, M2 z t => map_or (predta_coacc_0 d x z) (predta_coacc_0 d y t)
| _, _ => map_mini state d
end.
Definition predta_coacc (d : preDTA) (a : ad) (m : Map bool) :
Map bool := map_replace bool (predta_coacc_0 d d m) a true.
Definition predta_coacc_states (d : preDTA) (a : ad) :
Map bool :=
power (Map bool) (predta_coacc d a) (map_mini state d)
(S (MapCard state d)).
Definition predta_coacc_states_0 (d : preDTA) (a : ad) :
Map bool :=
lazy_power bool eqm_bool (predta_coacc d a) (map_mini state d)
(S (MapCard state d)).
(* lemmes sur map_or *)
(* lemmes sur map_replace : *)
(* définition ok de l'application predta_coacc *)
(* croissance de predta_coacc *)
Definition lemd (d : preDTA) : mRelation bool :=
fun m0 m1 : Map bool =>
ensemble_base state d m0 /\ ensemble_base state d m1 /\ lem m0 m1.
(* démonstration : propriété de chaine croissantes bornées *)
Definition lattice_lemd_bounded_0_def (p : prechain bool) : Prop :=
forall d : preDTA,
chain bool (ensemble_base state d) (lemd d) p ->
chain bool (ensemble_base state d) lem p.
Definition lattice_lemd_bounded_1_def (p : prechain bool) : Prop :=
lattice_lemd_bounded_0_def p ->
forall m : Map bool, lattice_lemd_bounded_0_def (concat bool p m).

Lemma lattice_lemd_bounded_3 :
forall (p : prechain bool) (d : preDTA),
chain bool (ensemble_base state d) (lemd d) p ->
chain bool (ensemble_base state d) lem p.