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Dec.v
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Dec.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 *)
(* *)
(****************************************************************************)
(* Dec.v *)
(****************************************************************************)
(* Formal Language Theory *)
(* *)
(* Judicael Courant - Jean-Christophe Filliatre *)
(* *)
(* Developped in V5.8 June-July 1993 *)
(* Ported to V5.10 October 1994 *)
(****************************************************************************)
Require Import Ensf.
Axiom Pdec : forall (P : Elt -> Prop) (x : Elt), {P x} + {~ P x}.
(* *)
(* Grace a l'axiome de decidabilite on peut construire, a partir d'un *)
(* ensemble e et d'un predicat f:Elt->Prop, le sous-ensemble des *)
(* elements de e qui verifient f. *)
(* *)
Fixpoint tq (f : Elt -> Prop) (e : Ensf) {struct e} : Ensf :=
match e return Ensf with
| empty =>
(* empty *) empty
(* add *)
| add x F =>
match Pdec f x return Ensf with
| left fx =>
(* true *) add x (tq f F)
(* false *)
| right nfx => tq f F
end
end.
(* *)
(* On montre maintenant que (tq f E) est bien l'ensemble voulu, *)
(* (ie) si x est dans (tq f E) alors il est dans E et (f x) est vrai...*)
(* *)
Lemma dans_tq_imp :
forall (x : Elt) (f : Elt -> Prop) (E : Ensf),
dans x (tq f E) -> dans x E /\ f x.
intros x f.
simple induction E.
replace (tq f empty) with empty; auto.
intro.
apply (dans_empty_imp_P x); auto.
intros a b H.
replace (tq f (add a b)) with
match Pdec f a return Ensf with
| left fa => add a (tq f b)
| right nfa => tq f b
end; auto.
elim (Pdec f a).
intros a0 H0.
cut (a = x :>Elt \/ dans x (tq f b)).
2: apply dans_add; auto.
intro H1; elim H1; clear H1.
intro H1; rewrite <- H1; auto.
intro.
cut (dans x b /\ f x); auto.
intro H2; elim H2; auto.
intros.
cut (dans x b /\ f x); auto.
intro H1; elim H1; auto.
Qed.
(* *)
(* ...et reciproquement si x est dans E et si (f x) est vrai alors *)
(* x est dans (tq f E). *)
(* *)
Lemma imp_dans_tq :
forall (x : Elt) (f : Elt -> Prop) (E : Ensf),
dans x E -> f x -> dans x (tq f E).
intros x f.
simple induction E.
intro.
apply (dans_empty_imp_P x); auto.
intros a b H H0 x0.
replace (tq f (add a b)) with
match Pdec f a return Ensf with
| left fa => add a (tq f b)
| right nfa => tq f b
end; auto.
elim (Pdec f a).
intro.
cut (a = x :>Elt \/ dans x b).
2: apply dans_add; auto.
intro H1; elim H1; clear H1.
intro H1; rewrite H1; auto.
auto.
intro.
cut (a = x :>Elt \/ dans x b).
2: apply dans_add; auto.
intro H1; elim H1; clear H1.
intro.
absurd (f a); auto.
rewrite H1; auto.
auto.
Qed.
(* *)
(* De dans_tq_imp on deduit facilement que (tq f a) est inclus *)
(* dans a. *)
(* *)
Lemma inclus_tq : forall (f : Elt -> Prop) (a : Ensf), inclus (tq f a) a.
unfold inclus in |- *.
intros.
cut (dans x a /\ f x); auto.
2: apply dans_tq_imp; auto.
intro H0; elim H0; auto.
Qed.