- This avoids confusion between `mnat` and `max_nat`. The `m` stands for `mono`.
- With `_mono` added, the `_auth` suffix in the algebra name no longer makes
  sense, so I removed it.
- This makes the names between the logic and the algebra-level library consistent.
- I also renamed `_frag` into `_lb` in the algebra-level library so as to make it
  consistent with the logic-level library.

Furthermore make the order of lemmas consistent and make the versions for the
fractions consistent.