(* 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                               *)
(*                                                                          *)
(****************************************************************************)
(*                                  Max.v                                   *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)
Require Export Arith.Max.
(*  On definit maintenant (sup x) pour un ensemble x, qui est		*)
(*    - soit O, si x ne contient pas d'entier				*)
(*    - soit (S n), si n est le plus grand entier de x			*)
(*									*)
(* Definition Z : Elt -> nat := [x:Elt] (<nat>Case x of (* natural *) [n:nat](S n) (* couple  *) [a:Elt][b:Elt]O (* up      *) [e:Ensf]O (* word    *) [w:Word]O end ). *)
Definition Z (x : Elt) : nat := match x with
| natural n => S n
| _ => 0
end.
(* Fixpoint sup [e:Ensf] : nat := (<nat>Case e of (* empty *) O (* add   *) [x:Elt][f:Ensf](max (Z x) (sup f)) end ). *)
Fixpoint sup (e : Ensf) : nat :=
match e with
| empty => 0
| add x f => max (Z x) (sup f)
end.
(*  Par definition on a :  *)

(* Specifications: *)
Lemma sup_add :
forall (x : Elt) (e : Ensf), sup (add x e) = max (Z x) (sup e) :>nat.