(* 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                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_types.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)


(*  On definit 3 "types" mutuellement inductifs : Elt, Ensf et Word      *)
(*  On distingue elt et mot, car on a besoin du type mot plus tard.      *)
(*  Les constructeurs up et word permettent repectivement de considerer  *)
(*  un ensemble ou un mot en tant qu'element.				 *)


Inductive Ensf : Set :=
  | empty : Ensf
  | add : Elt -> Ensf -> Ensf
with Elt : Set :=
  | natural : nat -> Elt
  | couple : Elt -> Elt -> Elt
  | up : Ensf -> Elt
  | word : Word -> Elt
with Word : Set :=
  | nil : Word
  | cons : Elt -> Word -> Word.(* 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                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_types.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)


(*  On definit 3 "types" mutuellement inductifs : Elt, Ensf et Word      *)
(*  On distingue elt et mot, car on a besoin du type mot plus tard.      *)
(*  Les constructeurs up et word permettent repectivement de considerer  *)
(*  un ensemble ou un mot en tant qu'element.				 *)


Inductive Ensf : Set :=
  | empty : Ensf
  | add : Elt -> Ensf -> Ensf
with Elt : Set :=
  | natural : nat -> Elt
  | couple : Elt -> Elt -> Elt
  | up : Ensf -> Elt
  | word : Word -> Elt
with Word : Set :=
  | nil : Word
  | cons : Elt -> Word -> Word.(* 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                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_types.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)


(*  On definit 3 "types" mutuellement inductifs : Elt, Ensf et Word      *)
(*  On distingue elt et mot, car on a besoin du type mot plus tard.      *)
(*  Les constructeurs up et word permettent repectivement de considerer  *)
(*  un ensemble ou un mot en tant qu'element.				 *)


Inductive Ensf : Set :=
  | empty : Ensf
  | add : Elt -> Ensf -> Ensf
with Elt : Set :=
  | natural : nat -> Elt
  | couple : Elt -> Elt -> Elt
  | up : Ensf -> Elt
  | word : Word -> Elt
with Word : Set :=
  | nil : Word
  | cons : Elt -> Word -> Word.(* 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                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_types.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)


(*  On definit 3 "types" mutuellement inductifs : Elt, Ensf et Word      *)
(*  On distingue elt et mot, car on a besoin du type mot plus tard.      *)
(*  Les constructeurs up et word permettent repectivement de considerer  *)
(*  un ensemble ou un mot en tant qu'element.				 *)


Inductive Ensf : Set :=
  | empty : Ensf
  | add : Elt -> Ensf -> Ensf
with Elt : Set :=
  | natural : nat -> Elt
  | couple : Elt -> Elt -> Elt
  | up : Ensf -> Elt
  | word : Word -> Elt
with Word : Set :=
  | nil : Word
  | cons : Elt -> Word -> Word.(* 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                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_types.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)


(*  On definit 3 "types" mutuellement inductifs : Elt, Ensf et Word      *)
(*  On distingue elt et mot, car on a besoin du type mot plus tard.      *)
(*  Les constructeurs up et word permettent repectivement de considerer  *)
(*  un ensemble ou un mot en tant qu'element.				 *)


Inductive Ensf : Set :=
  | empty : Ensf
  | add : Elt -> Ensf -> Ensf
with Elt : Set :=
  | natural : nat -> Elt
  | couple : Elt -> Elt -> Elt
  | up : Ensf -> Elt
  | word : Word -> Elt
with Word : Set :=
  | nil : Word
  | cons : Elt -> Word -> Word.(* 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                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_types.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)


(*  On definit 3 "types" mutuellement inductifs : Elt, Ensf et Word      *)
(*  On distingue elt et mot, car on a besoin du type mot plus tard.      *)
(*  Les constructeurs up et word permettent repectivement de considerer  *)
(*  un ensemble ou un mot en tant qu'element.				 *)


Inductive Ensf : Set :=
  | empty : Ensf
  | add : Elt -> Ensf -> Ensf
with Elt : Set :=
  | natural : nat -> Elt
  | couple : Elt -> Elt -> Elt
  | up : Ensf -> Elt
  | word : Word -> Elt
with Word : Set :=
  | nil : Word
  | cons : Elt -> Word -> Word.(* 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                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_types.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)


(*  On definit 3 "types" mutuellement inductifs : Elt, Ensf et Word      *)
(*  On distingue elt et mot, car on a besoin du type mot plus tard.      *)
(*  Les constructeurs up et word permettent repectivement de considerer  *)
(*  un ensemble ou un mot en tant qu'element.				 *)


Inductive Ensf : Set :=
  | empty : Ensf
  | add : Elt -> Ensf -> Ensf
with Elt : Set :=
  | natural : nat -> Elt
  | couple : Elt -> Elt -> Elt
  | up : Ensf -> Elt
  | word : Word -> Elt
with Word : Set :=
  | nil : Word
  | cons : Elt -> Word -> Word.(* 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                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_types.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)


(*  On definit 3 "types" mutuellement inductifs : Elt, Ensf et Word      *)
(*  On distingue elt et mot, car on a besoin du type mot plus tard.      *)
(*  Les constructeurs up et word permettent repectivement de considerer  *)
(*  un ensemble ou un mot en tant qu'element.				 *)


Inductive Ensf : Set :=
  | empty : Ensf
  | add : Elt -> Ensf -> Ensf
with Elt : Set :=
  | natural : nat -> Elt
  | couple : Elt -> Elt -> Elt
  | up : Ensf -> Elt
  | word : Word -> Elt
with Word : Set :=
  | nil : Word
  | cons : Elt -> Word -> Word.(* 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                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_types.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)


(*  On definit 3 "types" mutuellement inductifs : Elt, Ensf et Word      *)
(*  On distingue elt et mot, car on a besoin du type mot plus tard.      *)
(*  Les constructeurs up et word permettent repectivement de considerer  *)
(*  un ensemble ou un mot en tant qu'element.				 *)


Inductive Ensf : Set :=
  | empty : Ensf
  | add : Elt -> Ensf -> Ensf
with Elt : Set :=
  | natural : nat -> Elt
  | couple : Elt -> Elt -> Elt
  | up : Ensf -> Elt
  | word : Word -> Elt
with Word : Set :=
  | nil : Word
  | cons : Elt -> Word -> Word.(* 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                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_types.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)


(*  On definit 3 "types" mutuellement inductifs : Elt, Ensf et Word      *)
(*  On distingue elt et mot, car on a besoin du type mot plus tard.      *)
(*  Les constructeurs up et word permettent repectivement de considerer  *)
(*  un ensemble ou un mot en tant qu'element.				 *)


Inductive Ensf : Set :=
  | empty : Ensf
  | add : Elt -> Ensf -> Ensf
with Elt : Set :=
  | natural : nat -> Elt
  | couple : Elt -> Elt -> Elt
  | up : Ensf -> Elt
  | word : Word -> Elt
with Word : Set :=
  | nil : Word
  | cons : Elt -> Word -> Word.(* 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                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_types.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)


(*  On definit 3 "types" mutuellement inductifs : Elt, Ensf et Word      *)
(*  On distingue elt et mot, car on a besoin du type mot plus tard.      *)
(*  Les constructeurs up et word permettent repectivement de considerer  *)
(*  un ensemble ou un mot en tant qu'element.				 *)


Inductive Ensf : Set :=
  | empty : Ensf
  | add : Elt -> Ensf -> Ensf
with Elt : Set :=
  | natural : nat -> Elt
  | couple : Elt -> Elt -> Elt
  | up : Ensf -> Elt
  | word : Word -> Elt
with Word : Set :=
  | nil : Word
  | cons : Elt -> Word -> Word.(*  Inversion de quelques constructeurs...  *)
(*
Definition natural_inv : Elt -> nat :=
  [e:Elt]
    (<nat>Case e of
        (* natural *) [n:nat]n
        (* couple  *) [a:Elt][b:Elt]O
	(* up      *) [e:Ensf]O
	(* word    *) [w:Word]O
    end ).
*)
Definition natural_inv (e : Elt) : nat :=
  match e with
  | natural n => n
  | _ => 0
  end.

(* Specifications: *)
Lemma nat_invol : forall n : nat, natural_inv (natural n) = n.