(* 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            *)
(*                       COQ v5.10                              *)
(*                                                              *)
(* Laurent Arditi.  Laboratoire I3S. CNRS ura 1376.             *)
(* Universite de Nice - Sophia Antipolis                        *)
(* arditi@unice.fr, http://wwwi3s.unice.fr/~arditi/lolo.html    *)
(*                                                              *)
(* date: july 1995                                              *)
(* file: Definitions.v                                          *)
(* contents: PROOF OF A SEQUENTIAL MULTIPLIER. Definitions of   *)
(* auxilliary functions, axioms and the register-functions.     *)
(****************************************************************)
(****************************************************************)
(* Definitions des operateurs utilises (lowbit et highs) des registres (R1 et R2) et des lemmes sur les longueurs des registres *)
Require Export BV.
Require Export Adder.
Definition Mux := If BV.
Hint Unfold Mux.
Definition lowbit (l : list bool) :=
match l with
| nil => false
| b :: _ => b
end.
Definition highs (l : list bool) :=
match l with
| nil => nil (A:=bool)
| _ :: v => v
end.
simple induction v. intro. absurd (nil <> nilbv); unfold not in |- *; auto with arith.
intros. simpl in |- *. rewrite (trunc_O bool). trivial with arith.
Qed.
intros. unfold abit in |- *. unfold elemlist in |- *. rewrite strip_O.
apply lowbit_is_trunc. exact H.
Qed.
simple induction v. simpl in |- *. auto with arith.
intros. simpl in |- *. rewrite (strip_cons_S bool). rewrite strip_O. trivial with arith.
Qed.

Lemma app_lowbit_highs :
forall v : BV, v <> nilbv -> appbv (consbv (lowbit v) nilbv) (highs v) = v.