mathlib4

ng work (like business software), I'd be surprised if there were any useful programs that can't be boiled down to `while True: f(x)` where f(x) is some finite terminating program that you can prove stuff about. When you do a bunch of work in a language/ecosystem like Scala's, you get used to the idea that if your function says it returns an A, then it returns an A; there's generally no failure modes to consider if they're not in the type (ignoring realities like a datacenter catching on fire that are outside of the programming model). Passing a B to a `Function[B,A]` is an actual proof that you now have an `A` in your hand.

s it harder to learn the language (and, in particular, translate other ways of expressing mathematics into TLA+). I would love to see somebody who thinks deeply about PL syntax to make another language with the same semantics as TLA+, the same goals of looking like mathematics, but more familiar syntax. I don’t know what that would look like, but I’d love to see it. It seems like with the right library (see my mathlib point) and syntax, writing a TLA+ program should be no harder than writing a P program for the same behavior, but that’s not where we are right now. > The errors [types] catch are almost always quickly found by model checking. This hasn’t been my experience, and in fact a lot of the TLA+ programs I see contain partial implementations of arbitrary type checkers. I don’t think TLA+ needs a type system like Coq’s or Lean’s or Haskell’s, but I do think that some level of type enforcement would help avoid whole classes of common specification bugs (or even auto-generation of a type checking specification, which may be the way to go). > [A Coq-like type system] would put TLA+ beyond the ability of so many potential users that no proposal to add them should be taken seriously. I do think this is right, though. > This may turn out to be unnecessary if provers become smarter, which should be possible with the use of AI. Almost definitely will. This just seems like a no-brainer to bet on at this stage. See AlphaProof, moogle.ai, and many other similar examples. > A Unicode representation that can be automatically converted to the ascii version is the best alternative for now. Yes, please! Lean has a unicode representation, along with a nice UI for adding the Unicode operators in VSCode, and it’s awesome. The ASCII encoding is still something I trip over in TLA+, even after a decade of using it.

range of theories including Zermelo set theory near the lower end of the scale and going up to ZFC extended with large cardinal hypotheses such as "there is a measurable cardinal"; and on the other hand a hierarchy of extensions of NFU which is surveyed in the New Foundations article. These correspond to different general views of what the set-theoretical universe is like IEEE-754 specifies that float64s have ±infinity and specify ZeroDivisionError. Symbolic CAS with MPFR needn't be limited to float64s. HoTT in CoQ: Coq-HoTT: https://github.com/HoTT/Coq-HoTT What is the relation between Coq-HoTT & Homotopy Type Theory and Set Theory with e.g. ZFC? Homotopy Type Theory: https://en.wikipedia.org/wiki/Homotopy_type_theory : > "Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities" (2018) https://scholar.google.com/scholar?cites=9763697449338277760... and sorted by date: https://scholar.google.com/scholar?hl=en&as_sdt=5,43&sciodt=... What does Mathlib have for SetTheory, ZFC, NFU, and HoTT? leanprover-community/mathlib4// Mathlib/SetTheory: https://github.com/leanprover-community/mathlib4/tree/master...