Definition N_larger_bound n b := match n with
| 0 => larger_bound xH b
| Npos p0 => larger_bound p0 b
end.
Fixpoint lower_bound p0 := match p0 with
| xH => xH
| xO p0 => xO (lower_bound p0)
| xI p0 => xO (lower_bound p0)
end.

Lemma lower_bound_le p0 p1 :
(p0 <= p1 -> {k | lower_bound p1 = k * lower_bound p0})%positive.