intros. rewrite R1_eq2. rewrite (length_app bool).
unfold Mux in |- *. rewrite (F_If BV nat _ _ _ (@length bool)). simpl in |- *. rewrite If_eq.
rewrite highs_is_strip. rewrite (length_strip bool). unfold lengthbv in H.
rewrite H. symmetry  in |- *. rewrite plus_comm. apply le_plus_minus. auto with arith.
apply le_Sn_le; auto with arith.
unfold lengthbv in H. rewrite H. auto with arith.
apply le_Sn_le. exact H0.
Qed. Hint Resolve length_R1.

Lemma length_R2 : forall t : nat, t <= size -> lengthbv (R2 t) = size.
simple induction t. simpl in |- *.