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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                                                     *)
(** %\subsection*{ support :  empty.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export seq_set.
(** - Lemmas on empty sets and sequences *)
(* The empty sequences: *)
(** %\label{emptyseq}% *)
Definition empty_seq : forall A : Setoid, seq 0 A.
intro.
apply
(Build_Map
(Ap:=fun i : fin 0 =>
match i with
| Build_finiteT n Hn => False_rect A (lt_n_O n Hn)
end)); auto with algebra.
red in |- *.
intros.
apply False_ind; auto with algebra.
Defined.
intros.
simpl in |- *.
red in |- *.
intros.
apply False_ind; auto with algebra.
Qed.
Hint Resolve seq_O_is_empty: algebra.
intros.
apply Trans with (seq_set (empty_seq A)); auto with algebra.
simpl in |- *.
red in |- *.
split.
intro.
simpl in H.
inversion H; (apply False_ind; auto with algebra).
intro.
simpl in H.
contradiction.
Qed.
Hint Resolve seq_set_empty_seq_is_empty: algebra.

(* The only sequence of empty-set-elements is the empty sequence *)
Lemma no_seq_n_empty :
forall (n : Nat) (A : Setoid) (W : part_set A),
W =' empty A in _ -> seq n W -> n =' 0 in _.