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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 *)

Require Import Bool.
Require Import Words.

(****************)
(* 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).

Lemma paired_odd_l_intro :
 forall (b : bool) (w : word), paired w -> paired_odd_l b (bit b w).
Proof.
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_l (negb b) w), paired (bit b w) *)
unfold paired_odd_l in |- *; intros.
(* Goal: paired (bit (negb (negb b)) (bit (negb b) (append w0 (single (negb (negb b)))))) *)
apply paired_bit; trivial.
Qed.

Lemma paired_odd_l_elim :
 forall (b : bool) (w : word), paired_odd_l (negb b) w -> paired (bit b w).
Proof.
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_l (negb b) w), paired (bit b w) *)
unfold paired_odd_l in |- *; intros.
(* Goal: paired (bit b w) *)
rewrite (negb_intro b); trivial.
Qed.

(* 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).

Lemma paired_odd_r_from_rot :
 forall (w : word) (b : bool),
 paired_rot b w -> paired_odd_r b (bit (negb b) w).
Proof.
(* Goal: forall (w : word) (b : bool) (_ : paired_rot b w), paired (rotate w) *)
simple induction 1.
(* Goal: forall b : bool, paired_odd_r b (bit (negb b) empty) *)
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired_odd_r b (bit (negb b) (bit b w)) *)
intro; unfold paired_odd_r in |- *; simpl in |- *.
(* Goal: paired (bit (negb b0) (single b0)) *)
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired_odd_r b (bit (negb b) (bit b w)) *)
unfold single in |- *; apply paired_bit.
(* Goal: paired empty *)
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired_odd_r b (bit (negb b) (bit b w)) *)
apply paired_empty.
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired_odd_r b (bit (negb b) (bit b w)) *)
intros b0 b' w'; unfold paired_odd_r in |- *; intros.
(* Goal: paired (append (bit (negb b0) (bit b0 b')) (single b0)) *)
simpl in |- *; apply paired_bit; auto.
Qed.

(* 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).
Proof.
(* Goal: forall (b : bool) (w : word) (_ : paired_bet (negb b) w), paired_odd_r b (bit b w) *)
intros b w pb.
(* Goal: paired_odd_r b (bit b w) *)
rewrite (negb_intro b).
(* Goal: paired_odd_r (negb (negb b)) (bit (negb (negb b)) w) *)
elim pb.
(* Goal: forall (w : word) (_ : paired_odd_r (negb (negb b)) w), paired_odd_r (negb (negb b)) (bit (negb (negb b)) (bit (negb b) w)) *)
unfold paired_odd_r in |- *. (* Unfolds twice *)
(* Goal: forall (w : word) (_ : paired (append w (single (negb (negb b))))), paired (append (bit (negb (negb b)) (bit (negb b) w)) (single (negb (negb b)))) *)
intros; simpl in |- *.
(* Goal: paired (bit (negb (negb b)) (bit (negb b) (append w0 (single (negb (negb b)))))) *)
apply paired_bit; trivial.
Qed.


(************)
(* Rotation *)
(************)

Definition rotate (w : word) : word :=
  match w with
  | empty => empty
  | bit b w => append w (single b)
  end.

Lemma paired_rotate :
 forall (w : word) (b : bool), paired_rot b w -> paired (rotate w).
Proof.
(* Goal: forall (w : word) (b : bool) (_ : paired_rot b w), paired (rotate w) *)
simple induction 1.
(* Goal: paired empty *)
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired_odd_r b (bit (negb b) (bit b w)) *)
intro; simpl in |- *; apply paired_empty.
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired (rotate (bit b w)) *)
intros b' w'; simpl in |- *.
(* Goal: forall _ : paired_odd_r b' w', paired (append w' (single b')) *)
unfold paired_odd_r in |- *; trivial.
Qed.
