Long division verified via Hoare logic

Much like you have to cope with hash or GUID collisions... that is, you don't, because it statistically never happens. Unless you're speedrunning super mario or something.

Besides if you have a program that's formally verified, you just need to do what NASA did for its Apollo missions and make all the logic redundant and gate it behind a consensus algorithm.

You can argue that all 4 computers might get hit by a cosmic ray in just the right place and at just the right time... But it will never ever happen in the history of ever.

So my point is that the real world is messy. But the systems we design as engineers are not necessarily as messy. And the interface between the real world and our systems can be managed, and the proof of it is that we wouldn't be chatting across an entire ocean by modulating light itself if that weren't the case.

OK, but given that you don't know what that requirement is, how does that help you?

var x: int = y + 5

Here x is the variable being defined, it is declared to hold values of type int, and its initial value is given by the term y + 5.

In many mainstream languages, types and terms live in distinct universes. One starts by asking whether types and terms are all that different. The first step in this direction of inquiry is what are called refinement types. With our imaginary syntax, you can write something like:

val x: { int | _ >= 0 } = y + 5

Once again, x is the variable being defined, it is declared to always hold a value of type int at all relevant instants in all executions, and that its initial value is given by the term y + 5. But we additionally promise that x will always hold a non-negative value, _ >= 0. For this to typecheck, the typechecker must somehow also confirm that y + 5 >= 0.

But anyway, we have added terms to the previously boring world of types. This allows you to do many things, like so:

val x: int = ... val y: int = ... val z: { int | _ >= x && _ >= y } = if x >= y then x else y

We not only declare that z is an integer, but also that it always holds a value that exceeds both x and y.

You asked for the type of a function that multiplies two numbers. The type would look weird, so let me show you an imaginary example of the type of a function that computes the maximum:

val f : (x : int) -> (y : int) -> { int | _ >= x && _ >= y } = ...

This doesn't really get you to the maximum, because f might be computing max(x, y) + 5, but it does show the idea.

The final step in this direction is what are called full-blown dependent types, where the line between types and terms is completely erased.

That isn't what people are talking about when they talk about formal verification with a type system. They are talking about much more complex type systems that have equivalent power and expressivity to formal logic.

This intro is good at introducing the topic. Unfortunately, the correspondence between advanced type systems and logic is itself fairly abstract and uses lots of concepts in non-intuitive ways. This means that you can easily wind up in a situation where you can do lots of proofs about number theory using a tool like Lean4, but you don't really see the corresponding types (this is what happened to me).

https://leanprover-community.github.io/learn.html

These ideas have not caught on for a reason. You end up writing the logic twice, and pretty high chances you mess up once.