[markusde]

The examples in this book are extraordinarily simple, and covers material that many proof assistants were designed to be extremely good at expressing. I wouldn't be surprised if a LLM could automate the exercises in this book completely.

Writing nontrivial proofs in a theorem prover is a different beast. In my experience (as someone who writes mechanized mathematical proofs for a living) you need to not only know the proof very well beforehand, but you also need to know the design considerations for all of the steps you are going to use beforehand, and you also need to think about all of the ways your proof is going to be used beforehand. Getting these wrong frequently means redoing a ton of work, because design errors in proof systems are subtle and can remain latent for a long time.

[red_trumpet]

> think about all of the ways your proof is going to be used beforehand

What do you mean by that? I don't know much about theorem provers, but my POV would be that a proof is used to verify a statement. What other uses are there one should consider?

[markusde]

The issue is-- there are lots of way to write down a statement.

One common example is if you're going to internalize or externalize a property of a data structure: eg. represent it with a dependent type, or a property about a non-dependent type. This comes with design tradeoffs: some lemmas might expect internalized representations only, some rewrites might only be usable (eg. no horrifying dependent type errors) with externalized representations. For math in particular, which involves rich hierarchies of data structures, your choice about internalization might can impacts about what structures from your mathematical library you can use, or the level of fragile type coercion magic that needs to happen behind the scenes.