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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: may 1995                                               *)
(* file: AdderProof                                             *)
(* contents: verification of the adder defined in Adder.v       *)
(*                                                              *)
(****************************************************************)
Require Export Adder.
(****************************************************************)
unfold nilbv in |- *. simple induction v; auto with arith. unfold BV_full_adder in |- *. intros.
rewrite BV_full_adder_sum_eq3. rewrite BV_full_adder_carry_eq3.
rewrite (app_eq2 bool). rewrite half_adder_carry_true.
simpl in |- *. elim a. unfold appbv in H. rewrite H. simpl in |- *. auto with arith.
rewrite BV_full_adder_sum_v_nil_false.
rewrite BV_full_adder_carry_v_nil_false. rewrite BV_to_nat_app2.
simpl in |- *. elim mult_n_O. elim plus_n_O. trivial with arith.
Qed.
simple induction v. simple induction cin; auto with arith.
simple induction cin. rewrite BV_full_adder_nil_true_ok. simpl in |- *. auto with arith.
unfold BV_full_adder in |- *. rewrite BV_full_adder_sum_v_nil_false.
rewrite BV_full_adder_carry_v_nil_false. rewrite BV_to_nat_app2.
simpl in |- *. elim mult_n_O. elim plus_n_O. trivial with arith.
Qed.
(****************************************************************)

Theorem BV_full_adder_ok :
forall (v w : BV) (cin : bool),
BV_to_nat (BV_full_adder v w cin) =
BV_to_nat v + BV_to_nat w + bool_to_nat cin.