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Max.v
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(* 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 Import Le.
Require Import Lt.
Require Import Ensf.
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 : *)
Lemma sup_add :
forall (x : Elt) (e : Ensf), sup (add x e) = max (Z x) (sup e) :>nat.
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.
*)
Lemma elt_not_sym : forall a b : Elt, a <> b :>Elt -> b <> a :>Elt.
auto.
Qed.
(* (Z (natural n)) vaut (S n), donc est plus grand que n *)
Lemma lt_n_Z : forall n : nat, n < Z (natural 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) *)
(* *)
Lemma lt_n_sup : forall (x : Ensf) (n : nat), dans (natural n) x -> n < 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 *)
(* *)
Lemma sup_out : forall x : Ensf, ~ dans (natural (sup x)) 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.
intro.
exists (natural (sup e)).
apply sup_out.
Qed.