mm0

That is basically the pitch for Metamath Zero[0]. I'm philosophically aligned with you. Basically, I think this is because of where resources are invested - there's a tremendous amount of work on programming languages to make them ergonomic and enable composition of pieces. Much of that work is at its essence processing friendly syntax into some form of typed lambda calculus. Since dependently typed lambda calculus is a very handy way to encode proofs, it's easier to just adapt that all that mechanism than to develop infrastructure just for proofs. Interop is hard, because all the axiom systems tend to be subtly different in ways that are hard for normal people to appreciate, though there is some work on it, including MM0. [0]: https://github.com/digama0/mm0

Perhaps look into Metamath Zero / mm0 which.. well I'll just quote from the project [1]: > Metamath Zero is a language for writing specifications and proofs. Its emphasis is on balancing simplicity of verification and human readability of the specification. That is, it should be easy to see what exactly is the meaning of a proven theorem, but at the same time the language is as pared down as possible to minimize the number of complications in potential verifiers. > The goal of this project is to build a formally verified (in MM0) verifier for MM0, down to the hardware, to build a strong trust base on which to build verifiers. [1]: https://github.com/digama0/mm0

IMO an interesting project in this space is: mm0 / MetaMath Zero - Closing the loop in proof verification down to verifying the machine code of the verifier. Goes from first-order logic to peano arithmetic to a model of x86 to a model of the verifier written in x86. Interestingly, it demonstrates that verification of a compact proof can be performed in linear time (!) if the proof is structured correctly. -- https://github.com/digama0/mm0 The fact that proof checking can take linear time (though not proof-finding), and the fact that it incorporates so many 'layers' has emboldened my opinion that such a thing as I described above is possible and has enormous potential.