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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                               *)
(*                                                                          *)
(****************************************************************************)
(*                                  gram.v                                  *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)
Definition Mots (X : Ensf) :=
forall a : Elt, dans a X -> exists w : Word, word w = a.
Definition Regles (X V R : Ensf) :=
forall x : Elt,
dans x R ->
ex2 (fun A : Elt => dans A V)
(fun A : Elt =>
ex2 (fun B : Word => x = couple A (word B))
(fun B : Word => inmonoid (union X V) B)).
intros X V R x y Regles_R dans_couple_R.
cut
(ex2 (fun A : Elt => dans A V)
(fun A : Elt =>
ex2 (fun B : Word => couple x y = couple A (word B))
(fun B : Word => inmonoid (union X V) B))).
intro temp; elim temp; clear temp.
intros x0 dans_x0_V temp; elim temp; clear temp.
intros u eg_couple inmonoid_u.
replace x with x0; prolog [ sym_equal couple_couple_inv1 ] 3.
(*Assumption. Apply sym_equal. Apply couple_couple_inv1 with y (word u); Assumption.*)
auto.
Qed.

Lemma Regles_inv2 :
forall (X V R : Ensf) (x : Elt) (u : Word),
Regles X V R -> dans (couple x (word u)) R -> inmonoid (union X V) u.