[antpls]

> However, if you make a mistake in your proof, then the compiler will happily verify the wrong thing.

I don't use theorem provers in my to day to day tasks, but I had the 101 course at University. Back then I wrote basic proof with the tools, and if your problem is correctly stated and entered in the system, you simply CAN'T arrive to an invalid proof. The only way to fool the compiler is to use logical shortcuts, which are clearly defined in term of language keywords, so you know exactly where is the weakness in the proof, and look for them.

Edit : I don't know every theorem provers out there, so to give a bit more context about my experience, it was with the Coq theorem prover

[kazinator]

> if your problem is correctly stated and entered in the system, you simply CAN'T arrive to an invalid proof.

How is that different from: If your problem is correctly stated, and correctly coded into in the system using x86 machine language, then you can't arrive at a bug!

[antpls]

At least in Coq, there is no "bug" when you write a proof, it uses mathematical notations and logical rules to conclude a fact. Because of that you cannot state all the problems you would be able to state with x86. What I meant is that such theorem provers don't give you false-positive : if it tells you your proof is correct, then there can't be a "bug" in your reasoning

[yogthos]

In Coq your proof is your program, and if you end up proving the wrong thing that's a bug in the program. There's no magic here. Only a human can tell whether the code does what was intended. To do that you have to understand the code.

[antpls]

If you are proving the wrong statement, then you simply didn't state the initial problem correctly, it has nothing to do with a bug in your proof. It's like saying there is a bug in your implementation of a sort function, while we initially asked you to implement a max function. Even if your sort function is correctly implemented, it's still a "bug" to the person who asked you to implement a max function.

[yogthos]

That's my whole point. You're really just pushing the problem as step back instead of solving it. Instead of worrying about bugs in your implementation, you're now worrying about bugs in your specification.

[kazinator]

Problem is that these proofs are not the specification. They are very detailed. Specification says "write me a sort function", and the proof is some gobbledygook that deal with irrelevant minutiae of the sorting implementation. Where is the proof which proves that that proof matches the specification?

[antpls]

A theorem prover let you prove that your code is bug free in some cases. Why not use a theorem prover for your specifications too?