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Alternate.v
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(* 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 *)
Require Import Bool.
Require Import Words.
(*********************)
(* 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.
Lemma alt_neg_intro :
forall (b b' : bool) (w : word), alt b (bit b' w) -> alt (negb b) w.
Proof.
intros b b' w al.
inversion al; trivial.
Qed.
Lemma alt_neg_elim :
forall (b b' : bool) (w : word), alt (negb b) (bit b' w) -> alt b w.
Proof.
intros; rewrite (negb_intro b); apply alt_neg_intro with b'; trivial.
Qed.
Lemma alt_eq : forall (b b' : bool) (w : word), alt b (bit b' w) -> b = b'.
Proof.
intros b b' w al.
inversion al; trivial.
Qed.
Lemma alt_back :
forall (b b' : bool) (w : word),
alt b (bit b' w) -> b = b' /\ alt (negb b) w.
Proof.
intros; split.
apply alt_eq with w; trivial.
apply alt_neg_intro with b'; trivial.
Qed.
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_l :
forall u v w : word, conc u v w -> forall b : bool, alt b w -> alt b u.
Proof.
simple induction 1; auto; intros.
elim alt_back with b0 b w0.
intros eq A.
elim eq; auto.
trivial.
Qed.
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.
Proof.
simple induction 1; intros.
right; split; intros; auto.
elim H1 with (negb b0).
rewrite (negb_elim b0); intro.
right; split; elim H3; auto.
intro; left; split; elim H3; auto.
apply alt_neg_intro with b; trivial.
Qed.
Lemma alt_conc :
forall u v w : word, conc u v w -> alternate w -> alternate u /\ alternate v.
Proof.
simple induction 2; intros b ab; split.
apply alter with b; apply alt_conc_l with v w; trivial.
elim alt_conc_r with u v w b; intros; trivial.
elim H1; intros; apply alter with (negb b); trivial.
elim H1; intros; apply alter with b; trivial.
Qed. (* unused in Shuffle *)