simple induction t. elim minus_n_O. rewrite <- (length_R1 0).
intro. rewrite (trunc_all bool). rewrite (strip_O bool). auto with arith.
auto with arith. intros. rewrite R1_eq2. rewrite highs_is_strip.
rewrite (trunc_app bool). rewrite (trunc_strip bool).
rewrite (length_strip bool). rewrite length_R1.
replace (size - S n - (size - 1)) with 0.
rewrite (trunc_O bool).
replace (1 + (size - S n)) with (size - n).
rewrite (app_v_nil bool). unfold truncbv in H.
rewrite H. rewrite (strip_strip bool). rewrite plus_comm. simpl in |- *. auto with arith.
apply le_Sn_le; exact H0.
simpl in |- *. rewrite minus_Sn_m. simpl in |- *. trivial with arith.
exact H0.
symmetry  in |- *. apply minus_le_O. apply le_minus_minus. auto with arith.
apply le_Sn_le; exact H0.
rewrite length_R1. apply le_SO_size. apply le_Sn_le; exact H0.
Qed.
(* Lemma R1_lem2 : (t,i:nat) (* R1(t)[i]=B[t+i] *)
(le t size) -> (lt i size)  ->
(abit (R1 t) i) = (abit v1 (plus t i)).
*)

(* Specifications: *)
Lemma R1_lem3 :
forall t : nat, (* R1(t)[0]=B[t] *)
t < size -> abit (R1 t) 0 = abit V1 t.