Those tests that you list later are "happy path" tests. We want to know that that works, but we can't rely on only that sort of test especially if the type system doesn't work with us to avoid incorrect inputs to the function.
Honestly if a sad path causes a typing error in elixir it's not the worst thing. Sentry will catch it, only the user thread crashes, and you go patch it later.
A type is literally a description of a set of valid values. So when you say you test with bad values, then the answer is: you could use types and would not need these tests anymore.
However, the more interesting question is: is your typesystem capable of expressing your type and, if so, is it worth the effort and implications to do so.
But on a more theoretical level, OP is right: you _can_ save the tests with, given a powerful enough typesystem.
No. These are not internal contracts, these are contracts with user input. In a statically typed language, You are still advised to write tests that your marshalling technique provides the expected error (and downstream effects) that you plan for, if say the user inputs the string "1.", For a string that should be marshalled as an integer.
> A type is literally a description of a set of valid values.
That is generally not the case. There are, for example cases where certain floating points are invalid inputs (either outside of the domain of the function or a singular points), and I don't know of a mainstream PL that lets you define subsets of the reals in their type system.
In go, or c, c++, or rust, you could have a situation where a subset of integers are valid values (perhaps you need "positive, nonzero integers", because you are going to do an integer divide at some point) and that is not an expressible set of value in that type system. Ironically, that is a scenario that IS typable in Elixir and erlang, which are dynamically typed languages.
Yes and no. The trick is building a type system simultaneously strong enough to encode the properties you want, and weak enough that it's statically decidable.
There will always be properties you can't encode in a (useful) type system.
It's a fine line to walk, really. There's an argument to be made that most of the times we don't actually need Turing-completeness, and we'd better off using only types to encode computations, but OTOH I don't really want to think about what coinductive types I must define to solve a problem that could be solved with five lines of javascript instead.
If you define useful by "we know that the compiler will finish in finite time" then I agree. And that's indeed a good point! In practice, there will always be runtime tests, at least for how long any of us and our children will live. :)
Re your second code block: This can be typed with literal types, no need for tests at compile time.
It's a misconception that developers who use dynamically typed programming languages write tests that perform tasks of a static type system. They do not write tests like this:
They write tests like