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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                               *)
(*                                                                          *)
(****************************************************************************)
(*                                Ensf_map.v                                *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)
(*									*)
(*  MAP "a la CAML"							*)
(*    On definit une fonction map qui applique une fonction a tous les  *)
(*    elements d'un ensemble et renvoie l'ensemble des resultats	*)
(*    Ceci permet, entre autres, de definir facilement l'union		*)
(*    disjointe (Voir ci-dessous)					*)
(*									*)
Fixpoint map (f : Elt -> Elt) (e : Ensf) {struct e} : Ensf :=
match e with
| empty => empty
| add y e => add (f y) (map f e)
end.
(*									*)
(*  On montre ici le resultat general suivant :				*)
(*    (dans  x (map f A)) -> il existe y dans A tel que x=f y		*)
(*									*)

Lemma dans_map :
forall (f : Elt -> Elt) (a : Ensf) (x : Elt),
dans x (map f a) -> exists y : Elt, dans y a /\ x = f y.