Missing headers in leanc makes compiling certain FFI libraries painful

Prerequisites

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• Check that your issue is not already filed:
  https://github.com/leanprover/lean4/issues
• [ X] Reduce the issue to a minimal, self-contained, reproducible test case.
  Avoid dependencies to Mathlib or Batteries.
• [ X] Test your test case against the latest nightly release, for example on
  https://live.lean-lang.org/#project=lean-nightly
  (You can also use the settings there to switch to “Lean nightly”)

Description

Lean comes with a C compiler called leanc, since Lean itself compiles down to C itself.
This is really useful!
However, leanc is missing a bunch of important C headers like stdio.h, which makes compiling certain very important C libraries like SQLite or SDL impossible without gcc or clang.
It would be nice to be able to add the full C standard library to be able to compile more stuff.

Context

#general > SQLite bindings for Lean @ 💬

Steps to Reproduce

N/A

Versions

Lean 4.24, Windows 11

Additional Information

I'm currently working on SQLite bindings for Lean, and I worked on SDL bindings for lean, so in the past, I've had to rely on solutions like using MSYS2 or WSL to compile such bindings on Windows, which is very inconvenient. It would be nice not to have to worry about such things and use Lean's built in C compiler.

Impact

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Start with the leanc entry point and reproduce the missing stdio.h case on Windows 11; the issue names no source file or test. Completion means the needed C standard headers are available for the SQLite or SDL FFI use case without relying on MSYS2 or WSL, with a regression test covering the reported case.