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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                                 *)
(*                                                                          *)
(****************************************************************************)
(*                                 Paired.v                                 *)
(****************************************************************************)
(* G. Huet - V5.8  Nov. 1994 *)
(*    ported V5.10 June 1995 *)
(****************)
(* Paired words *)
(****************)
(* (paired w) ==  w = [b1 ~b1 b2 ~b2 ... bn ~bn] *)
Inductive paired : word -> Prop :=
| paired_empty : paired empty
| paired_bit :
forall w : word,
paired w -> forall b : bool, paired (bit (negb b) (bit b w)).
(* paired_odd_l b w == w = [b b1 ~b1 b2 ~b2 ... bn ~bn] *)
Definition paired_odd_l (b : bool) (w : word) := paired (bit (negb b) w).
(* paired_odd_r b w == w = [b1 ~b1 b2 ~b2 ... bn ~bn ~b] *)
Definition paired_odd_r (b : bool) (w : word) := paired (append w (single b)).
(* paired_rot b w == w = [b b2 ~b2 ... bn ~bn ~b] *)
Inductive paired_rot : bool -> word -> Prop :=
| paired_rot_empty : forall b : bool, paired_rot b empty
| paired_rot_bit :
forall (b : bool) (w : word),
paired_odd_r b w -> paired_rot b (bit b w).
(* paired_bet b w == w = [b b2 ~b2 ... bn ~bn b] *)
Inductive paired_bet (b : bool) : word -> Prop :=
paired_bet_bit :
forall w : word, paired_odd_r (negb b) w -> paired_bet b (bit b w).

Lemma paired_odd_r_from_bet :
forall (b : bool) (w : word),
paired_bet (negb b) w -> paired_odd_r b (bit b w).