(** The following axiom states that if a predicate P : Typ1->Prop holds for some type X : Typ1, then there exists a code c : Code such that P holds for (eps P c).  Technically, this axiom shows that we can always replace an existential statement over a large type (i.e. Typ1) by an existential statement over a small type (i.e. Code), which is crucial for interpreting the collection scheme. *)
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(** * An implementation of a super-powerset  *)
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(** On a fixed small type Z, we can build several pointed graphs of the form (Z, A, a), whose two extra parameters A and a range over (Rel Z) and Z respectively.  But since the cartesian product (Rel Z) * Z is itself a small type, it is not difficult to build a variant of the powerset (foo, FOO, foo_rt) that contains all the pointed graphs of the form (Z, A, a), that is, a pointed graph (called the "super-powerset" of Z) such that  (Z, A, a) \in (foo, FOO, foo_rt)  for any edge relation A : (Rel A) and for any root a : Z.  In what follows, we propose to achieve this construction, but in a more general setting.  For that, instead of considering a small type Z : Typ1, we will rather consider a family of small types Z : X->Y->Typ1 indexed by the inhabitants of two given small types X, Y : Typ1, and from these parameters, we propose to build a pointed graph ((foo X Y Z), (FOO X Y Z), (foo_rt X Y Z)) such that  ((Z x y), A, a) \in ((foo X Y Z), (FOO X Y Z), (foo_rt X Y Z))  for any x : X, y : Y, A : (Rel (Z x y)) and a : (Z x y). Intuitively, this pointed graph represents the union of the super-powersets associated to the carriers (Z x y) when x and y range over the small types X and Y respectively. *)
(** ** Building the carrier of the super-powerset **)
(** The carrier (foo X Y Z) of the super-powerset associated to a given family of small types Z : X->Y->Typ1 indexed by two small types X and Y would be naturally defined in Coq as follows:  Inductive foo [X,Y:Typ1; Z:X->Y->Typ1] : Typ1 := | foo_in : (x:X; y:Y)(Rel (Z x y))->(Z x y)->(foo X Y Z) | foo_rt : (foo X Y Z).  Intuitively, this type is the disjoint union of the cartesian products (Rel (Z x y)) * (Z x y) where x and y range over X and Y respectively, plus an extra element (foo_rt X Y Z) that will be used as the new root.  In our framework, this inductive definition is easily mimicked as follows: *)
Definition foo (X Y : Typ1) (Z : X -> Y -> Typ1) : Typ1 :=
(forall (x : X) (y : Y), Rel (Z x y) -> Z x y -> Prop) -> Prop.
Definition foo_in (X Y : Typ1) (Z : X -> Y -> Typ1)
(x : X) (y : Y) (R : Rel (Z x y)) (z : Z x y) : foo X Y Z :=
fun f => f x y R z.
Definition foo_rt (X Y : Typ1) (Z : X -> Y -> Typ1) :
foo X Y Z := fun _ => bot.
(** It is then easy to check the expected property of non-confusion holds for the constructors foo_in and foo_rt: *)

Lemma eq_foo_in_rt :
forall (X Y : Typ1) (Z : X -> Y -> Typ1) (x : X) (y : Y)
(R : Rel (Z x y)) (z : Z x y),
eq (foo X Y Z) (foo_in X Y Z x y R z) (foo_rt X Y Z) -> bot.