alt Hacker News(Asking as an interested noob) -- How is this different to something like 'assert' statements in Python?
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I’m an noob to Lean myself, but my best explanation is that asserts are checks that happen at runtime. Imagine compile-time asserts, where the compiler is able to determine the validity of assertions without needing to run the program. Formal verification techniques make this possible. In Lean, it is possible to write a proof that, if true, guarantees that the code works.
It works on the type system level instead of at runtime. So you don't actually need to "run" any code to verify it, and you can verify it for all possible inputs, even infinity of them, rather than for the ones that exist in your test.
Assertions are for testing at runtime. They demonstrate that the behavior is correct on one input when it runs. Formal verification proves that the code is correct on _all_ inputs _before_ it runs.
it's like proof by induction versus trying every number from 0 to infinity
it is very much a related idea. an `assert` statement in e.g. `python` is a statement your code is making about what it means to be correct, and furthermore a statement about what conditions would have to exist to validate the first statement: for example you might need to run it with certain inputs, on a certain file or kind of file.
`lean4` is very much about the same two ideas. you can make statements about what it means for the code to say something interesting, usually something relevant to whether or not it's correct, and you make statements about the circumstances in which you would evaluate that.
people are interested in `lean4` because it allows you to make more interesting statements of both kinds, and you have tools to be much more specific about the details, the `assert` statements in `python` can't really call each other for example, they don't really compose. in `lean4` the ability to compose such statements is very important.
but you can write regular programs in it too. this is a reverse proxy faster than `nginx`: https://cdn.s4.gl/serve-fd.lean
An assert statement requires that you specifically come up with a test case. Lean lets you verify for all possible cases. Infinity is not a problem.
It's similar to a type system in that regard. The same difference could be applied there (comparing a Python type assert). Types, however, generally only cover checks similar to "the shape of the data is X".
Lean is different in that its language for expressing properties is wide enough to express anything you can imagine. The bottleneck becomes accurately stating properties you'd like to enforce and, subsequently, discovering proofs of whether or not they're true.