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(* 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.
intros. rewrite lowbit_is_trunc. rewrite highs_is_strip.
rewrite (app_trunc_strip bool). trivial with arith. exact H.
Qed.
intros. rewrite highs_is_strip.
rewrite (length_strip bool). apply minus_n_SO.
apply (v_not_nil_length bool). exact H.
Qed.

Lemma length_abit :
forall (v : BV) (i : nat), i < lengthbv v -> lengthbv (abit v i) = 1.