isabelle-lambda-calculus VS cakeml

I got more into theorem proving, i.e. using a programming language to write mathematical proofs and have them checked by a compiler. Step by step I am working on a machine checked proof of type safety of System Fc, the intermediate language of the Haskell compiler.

If I understand what you are asking about correctly, then I do think you are mistaken.

As a sibling comment observed, you would be proving something about a program, but proving things about programs is both possible and done.

This ranges from things like CakeML (https://cakeml.org/) and CompCert (compilers with verified correctness proofs of their optimizations) to something simple like absence of runtime type errors in statically strongly soundly-typed languages.

Of note is that you are proving properties of your program, not proving them perfect in every way. The properties of your program that you prove can vary wildly in both difficulty and usefulness. A sufficiently advanced formally verified compiler like CakeML can transfer a high-level proof about your source code to a corresponding proof about the behavior of the generated machine-executable code.

> A single IR with multiple passes is a good way to build a compiler

https://mlir.llvm.org/, which is using, is largely claiming the opposite. Most passes more naturally are not "a -> a", but "a -> b". data structures and data structures work hand in hand, it is very nice to produce "evidence" for what is done in the output data structure.

This is why https://cakeml.org/, which "can't cheat" with partial functions, has so many IRs!

Using just a single IR was historically done for cost-control, the idea being that having many IRs was a disaster in repetitive boilerplate. MLIR seeks to solve that exact problem!

You may want to look at [CakeML](https://cakeml.org) done in HOL4, there is also a nice proof pearl about a more .. minimalistic verified bootstrapped compiler also in HOL4.

CakeML

There is also Cake ML which is a formally verified functional programming language compiler and runtime.

There is also a CakeML -> Standard ML compiler though it seems to have been built to translate benchmarks and sort of old so I'm not sure how comprehensive it is: https://github.com/CakeML/cakeml/tree/master/unverified/front-end

> One guess is that lisps cope with being minimal through use of macros and metaprogramming, it's difficult for a typed language to support that level of metaprogramming while maintaining the various guarantees that one wants from such a system.

Difficult, but certainly not impossible [0].

[0] https://cakeml.org/

If this interests you, I'd highly recommend checking out CompCert (docs here) and CakeML.