simple induction t. intros. elim minus_n_O. apply Invariant_t_O.
intros. rewrite <- (app_trunc_elemlist bool).
replace (elemlist bool V1 n) with (abit V1 n).
rewrite <- (R1_lem3 n). rewrite R1_eq2. rewrite R2_eq2.
rewrite <- lowbit_is_abit. case (lowbit (R1 n)).
replace (consbv (lowbit (BV_full_adder_sum (R2 n) V2 false)) nilbv) with
(abit (BV_full_adder_sum (R2 n) V2 false) 0).
apply inv_ind_true. auto with arith.
intros. apply H. exact H1. exact H0.
apply le_Sn_le; exact H0. unfold abit in |- *. rewrite lowbit_is_trunc.
unfold elemlist in |- *. rewrite (strip_O bool). trivial with arith.
apply (not_nil bool). rewrite length_BV_full_adder_sum. rewrite length_R2.
exact size_not_O. apply le_Sn_le; exact H0.
rewrite length_R2. rewrite length_V2_size. trivial with arith.
apply le_Sn_le; exact H0.
replace (consbv (lowbit (BV_full_adder_sum (R2 n) V2 false)) nilbv) with
(abit (BV_full_adder_sum (R2 n) V2 false) 0).
apply inv_ind_false. trivial with arith.
exact H. exact H0. apply le_Sn_le; exact H0. rewrite lowbit_is_trunc.
unfold abit in |- *. unfold elemlist in |- *. rewrite (strip_O bool). trivial with arith.
apply (not_nil bool).
rewrite length_BV_full_adder_sum. rewrite length_R2; auto with arith.
transitivity size.
rewrite length_R2; trivial with arith. apply le_Sn_le; exact H0. auto with arith.
apply (not_nil bool). rewrite length_R1; auto with arith. auto with arith. auto with arith.
replace (length V1) with (lengthbv V1). rewrite length_V1_size. exact H0.
auto with arith.
Qed.
(****************************************************************)
(* Et le theoreme final qui est une specialisation de l'invariant avec t=size *)

Theorem Correct :
BV_to_nat (appbv (R1 size) (R2 size)) = BV_to_nat V2 * BV_to_nat V1.