(* 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                                 *)
(*                                                                          *)
(****************************************************************************)
(*                               Alternate.v                                *)
(****************************************************************************)
(* G. Huet - V5.8  Nov. 1994 *)
(*    ported V5.10 June 1995 *)
(*********************)
(* Alternating words *)
(*********************)
(* (alt b w) == w = [b ~b b ~b ...]  *)
Inductive alt : bool -> word -> Prop :=
| alt_empty : forall b : bool, alt b empty
| alt_bit : forall (b : bool) (w : word), alt (negb b) w -> alt b (bit b w).
Hint Resolve alt_empty alt_bit.
Inductive alternate (w : word) : Prop :=
alter : forall b : bool, alt b w -> alternate w.
(* (alternate w) iff Exists b  (alt b w)  *)
(*********************************************)
(* Subwords of alternate words are alternate *)
(*********************************************)

Lemma alt_conc :
forall u v w : word, conc u v w -> alternate w -> alternate u /\ alternate v.