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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)).
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(** * The set of von Neuman numerals  *)
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(** The set omega of von Neumann numerals is represented as the pointed graph ((opt nat), OMEGA, (none nat)) whose edge relation OMEGA : (Rel (opt nat)) is defined by the following two clauses:  1. Delocation of the strict ordering "lt"  if (wf_nat n), (wf_nat m) and (lt n m), then (OMEGA (some nat n) (some nat m))  2. Connecting wf_nat's to the new root:  if (wf_nat n), then (OMEGA (some nat n) (none nat)).  As usual, we define it via a second-order encoding: *)
Definition OMEGA : Rel (opt nat) :=
fun z' z =>
forall E : Prop,
(forall n n' : nat,
eq (opt nat) z (some nat n) ->
eq (opt nat) z' (some nat n') -> wf_nat n -> wf_nat n' -> lt n' n -> E) ->
(forall n' : nat,
eq (opt nat) z (none nat) ->
eq (opt nat) z' (some nat n') -> wf_nat n' -> E) -> E.
(** The corresponding constructors are the following: *)
Proof
fun n Hn m Hm H E H1 _ =>
H1 m n (eq_refl (opt nat) (some nat m)) (eq_refl (opt nat) (some nat n)) Hm
Hn H.
Proof
fun n Hn E _ H2 =>
H2 n (eq_refl (opt nat) (none nat)) (eq_refl (opt nat) (some nat n)) Hn.
(** The pointed graph ((opt nat), OMEGA, (some nat O)) is bisimilar to the pointed graph (unit ZERO id) that represents the empty set. This is done by extensionality. *)

Lemma OMEGA_ZERO : EQV (opt nat) OMEGA (some nat O) unit ZERO id.