intuition.
unfold lognat.
eapply Nat_size_nat_monotonic.
specialize (Nnat.Nat2N.inj_compare n1 n2); intuition.
apply nat_compare_lt in H.
rewrite H in H0.
case_eq (N.of_nat n1 ?= N.of_nat n2)%N; intuition;
congruence.
Qed.

Lemma natToBv_bvToNat_inverse : forall n k,
(natToBv n (bvToNat k)) = k.