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(* as published by the Free Software Foundation; either version 2.1   *)
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(*                                                                    *)
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(* GNU General Public License for more details.                       *)
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(* You should have received a copy of the GNU Lesser General Public   *)
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(* 02110-1301 USA                                                     *)
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(** * Zero   *)
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(** 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.

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