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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                                 *)
(*                                                                          *)
(****************************************************************************)
(*                               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_r :
forall u v w : word,
conc u v w ->
forall b : bool, alt b w -> odd u /\ alt (negb b) v \/ even u /\ alt b v.