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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                               *)
(*                                                                          *)
(****************************************************************************)
(*                                  Rat.v                                   *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)
(*									*)
(*    On definit la concatenation, l'union, l'intersection de 2		*)
(*    langages, le langage L* si L est un langage, ainsi que le		*)
(*    langage reduit a un mot.						*)
(*									*)
Definition lword (w : Word) : wordset := fun w1 : Word => w = w1 :>Word.
Definition lconc (l1 l2 : wordset) : wordset :=
fun w : Word =>
exists w1 : Word,
(exists w2 : Word, l1 w1 /\ l2 w2 /\ w = Append w1 w2 :>Word).
Definition lunion (l1 l2 : wordset) : wordset := fun w : Word => l1 w \/ l2 w.
Definition linter (l1 l2 : wordset) : wordset := fun w : Word => l1 w /\ l2 w.
Fixpoint lpuiss (n : nat) : wordset -> wordset :=
fun l : wordset =>
match n return wordset with
| O =>
(* O *)  lword nil (* S *)
| S p => lconc l (lpuiss p l)
end.
Definition lstar (l : wordset) : wordset :=
fun w : Word => exists n : nat, lpuiss n l w.

Lemma induction_star :
forall (P : Word -> Prop) (l : wordset),
(forall (n : nat) (w : Word), lpuiss n l w -> P w) ->
forall w : Word, lstar l w -> P w.