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(* 02110-1301 USA                                                     *)


(*****************)
(** * Universes  *)
(*****************)

(** In what follows, we will work with explicit universes rather than
    with Coq-style implicit universes.

    For that, we define two universes Typ1 and Typ2 as shorthands for
    the generic universe Type by enforcing the constraint Typ1 : Typ2
    so that Coq knows that the implicit index associated to Typ1 is
    smaller than the implicit index associated to Typ2. *)

Definition Typ2 := Type.Definition Typ1 : Typ2 := Type.(* enforce Typ1 : Typ2 *)

(** Intuitively:

    - Typ1 is the universe of small types, which contains the type
      Prop of propositions, and is closed under dependent product.

    - Typ2 is the top-universe of large types, which contains the
      universe Typ1, and is closed under dependent product again.

    In the following, small types (i.e. the inhabitants of Typ1) will
    be used as the carriers of pointed graphs (for representing sets). *)

(** The type of binary relations: *)

Definition Rel (X : Typ1) : Typ1 := X -> X -> Prop.(***************************)
(** * Logical connectives  *)
(***************************)

(** To avoid the use of Coq-style inductive definitions, we redefine
    all the logical connectives from their second-order encoding and
    give their introduction rules.  Except in a few cases, it is not
    necessary to give the corresponding elimination rules whose
    behaviour is mimicked by using the tactic [Apply H] (instead of
    [Elim H]) in the subsequent proof-scripts. *)

(** ** Truth and falsity **)

Definition top : Prop := forall E : Prop, E -> E.

Lemma top_intro : top.