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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                               *)
(*                                                                          *)
(****************************************************************************)
(*                                 Words.v                                  *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)
Parameter alph : Ensf.
Parameter epsilon : Elt.
(*  On definit le predicat (inmonoid X w) qui signifie que le mot w	*)
(*  est dans le monoide libre engendre par X.				*)
Inductive inmonoid (X : Ensf) : Word -> Prop :=
| inmonoid_nil : inmonoid X nil
| inmonoid_cons :
forall (w : Word) (e : Elt),
inmonoid X w -> dans e X -> inmonoid X (cons e w).
Hint Resolve inmonoid_nil.
Hint Resolve inmonoid_cons.
(*  Inversion de la definition 						*)
(* Fixpoint Inmonoid [X:Ensf; w:Word] : Prop := (<Prop>Case w of (* nil  *) True (* cons *) [a:Elt][w':Word]( (dans a X) /\ (Inmonoid X w') ) end ). *)
Fixpoint Inmonoid (X : Ensf) (w : Word) {struct w} : Prop :=
match w with
| nil => True
| cons a w' => dans a X /\ Inmonoid X w'
end.
intros X w H.
elim H.
red in |- *; simpl in |- *; exact I.
intros.
change (dans e X /\ Inmonoid X w0) in |- *.
auto.
Qed.
Hint Resolve i_I.
intros X.
simple induction w.
auto.
intros x w0 H H0.
cut (dans x X /\ Inmonoid X w0); auto.
intro H1; elim H1; clear H1.
auto.
Qed.
Hint Resolve I_i.
intros.
cut (Inmonoid X w); auto.
cut (Inmonoid X (cons a w)); auto.
intro H0.
cut (dans a X /\ Inmonoid X w); auto.
intro H1; elim H1; clear H1.
auto.
Qed.
intros.
cut (Inmonoid X (cons a w)); auto.
intro.
cut (dans a X /\ Inmonoid X w); auto.
intro H1; elim H1; clear H1.
auto.
Qed.

Lemma inmonoid_inclus :
forall (E F : Ensf) (x : Word), inclus E F -> inmonoid E x -> inmonoid F x.