Nice, your antisymmetry proof turned out way nicer than mine did,
where I managed to miss add_zero/cancellation and wound up bottoming
out at n + ... \neq succ n + ...
I managed to get through quite a few, up to trichotomy. Except the
multiplication ones, which I could not figure out.
https://gist.github.com/ratmice/7df331c34ed5abfe6fe5e9d1c3d83636