(* This program is free software; you can redistribute it and/or      *)
(* 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.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* 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                                                     *)
(*************)
(** * Zero   *)
(*************)
(** In set theory, zero is implemented as the empty set.  As a pointed graph, we can take any (non-empty) carrier and any root, with an empty edge relation: *)
Definition unit : Typ1 := forall X : Typ0, X -> X.
Definition id : unit := fun X x => x.
Definition ZERO : Rel unit := fun _ _ => bot.

(* Specifications: *)
Lemma ZERO_elim :
forall (X : Typ1) (A : Rel X) (a : X), ELT X A a unit ZERO id -> bot.