(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(************************************************************************)
(* Copyright 2005 Milad Niqui                                           *)
(* This file is distributed under the terms of the                      *)
(* GNU Lesser General Public License Version 2.1                        *)
(************************************************************************)
(* $Id$ *)
(** We start by some general notions for expressing the bijection *)
Definition identity (A:Set) := fun a:A=> a.
Definition compose (A B C:Set) (g:B->C) (f:A->B) := fun a:A=>g(f a).
Section Denumerability.
(** What it means to have the same cardinality *)
Definition same_cardinality (A:Set) (B:Set) := {f:A->B & { g:B->A | (forall b,(compose _ _ _ f g) b= (identity B) b)
/\ forall a,(compose _ _ _ g f) a = (identity A) a}}.
(** What it means to be a denumerable *)
Definition is_denumerable A := same_cardinality A nat.

(* Specifications: *)
Lemma same_cardinality_transitive:forall A B C, same_cardinality A B -> same_cardinality B C -> same_cardinality A C.