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(* 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 :  *)
intros x.
simple induction e; auto.
Qed.
Hint Resolve sup_add.
(*  Finalement inutile :  *)
(* Lemma diff_natural : (n,m:nat)~(<nat>n=m)->~(<Elt>(natural n)=(natural m)). Intros; Red; Intro. Absurd (<nat>n=m). Assumption. Replace n with (natural_inv (natural n)). 2:Auto. Replace m with (natural_inv (natural m)). 2:Auto. Elim H0. Auto. Save. *)
(*  Finalement inutile  *)
(* Lemma lt_diff : (n,m:nat)(lt m n)->~(<nat>n=m). Intros. Red. Intro. Cut (lt m n); Auto. Elim H0. Change ~(lt n n). Auto. Save. *)
auto.
Qed.
(*  (Z (natural n)) vaut (S n), donc est plus grand que n		*)
intro.
replace (Z (natural n)) with (S n); auto.
Qed.
(*									*)
(*  On montre d'abord que tout entier dans x est strictement plus petit	*)
(*  que (sup x)								*)
(*									*)
simple induction x.
intros.
absurd (dans (natural n) empty); auto.
intros a b H n H0.
replace (sup (add a b)) with (max (Z a) (sup b)).
2: auto.
cut (n < Z a \/ n < sup b).
intro.
elim H1; auto.
intros; apply lt_le_trans with (Z a); auto with arith.
intros; apply lt_le_trans with (sup b); auto with arith.
cut (a = natural n :>Elt \/ dans (natural n) b).
2: apply dans_add; auto.
intro.
elim H1.
intro; left.
rewrite H2; apply lt_n_Z.
intro; right.
apply H; assumption.
Qed.
(*								*)
(*  On en deduit que (natural (sup x)) n'est pas dans x 	*)
(*								*)
intro.
red in |- *.
intro.
cut (sup x < sup x).
change (~ sup x < sup x) in |- *.
apply lt_irrefl.
apply lt_n_sup.
assumption.
Qed.
(*									   *)
(*  Le resultat final : 						   *)
(*      Pout tout ensemble e il existe un element x n'appartenant pas a e  *)
(*	(a savoir (natural (sup x)) )					   *)
(*									   *)

Lemma exist_other : forall e : Ensf, exists x : Elt, ~ dans x e.