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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.
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(** * The successor function   *)
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(** In set theory, the successor of von Neumann numeral n is encoded as the set  n U {n}, whose existence follows from the pairing axiom (used twice) and the union axiom, that have been already derived in the files "IZF_pair.v"  and "IZF_union.v".  However, it is simpler to give here a direct implementation of the successor as follows.  Formally, the successor of a given pointed graph (X, A, a) is represented as the pointed graph  ((opt X), (SUCC X A a), (none X))  whose edge relation is defined by the following three clauses:  1. Delocate A in the new graph via (some X):  if (A x' x), then (SUCC X A a (some X x') (some X x))  2. Connect the image of any direct element of the old root a:X to the new root:  if (A x a), then (SUCC X A a (some X x) (none X))  3. Connect the old root to the new root:  (SUCC X A a (some X a) (none X)).  As usual, we use a second-order encoding to define this relation. *)
Definition SUCC (X : Typ1) (A : Rel X) (a : X) : Rel (opt X) :=
fun z' z =>
forall E : Prop,
(forall x x' : X,
eq (opt X) z (some X x) -> eq (opt X) z' (some X x') -> A x' x -> E) ->
(forall x' : X,
eq (opt X) z (none X) -> eq (opt X) z' (some X x') -> A x' a -> E) ->
(eq (opt X) z (none X) -> eq (opt X) z' (some X a) -> E) -> E.
Proof
fun X A a x x' H E H1 _ _ =>
H1 x x' (eq_refl (opt X) (some X x)) (eq_refl (opt X) (some X x')) H.
Proof
fun X A a x' H E _ H2 _ =>
H2 x' (eq_refl (opt X) (none X)) (eq_refl (opt X) (some X x')) H.
Proof
fun X A a E _ _ H3 =>
H3 (eq_refl (opt X) (none X)) (eq_refl (opt X) (some X a)).

Lemma SUCC_eqv :
forall (X : Typ1) (A : Rel X) (a x : X),
EQV X A x (opt X) (SUCC X A a) (some X x).