[practal]

As I said, it depends on how you practically implement it.

I've used it for proving linear inequalities as part of the Flyspeck project (formal proof of the Kepler conjecture), and there I implemented my own rewrite engine for taking a set of rewrite rules and do the computation outside of the LCF kernel, for example by compiling the rules to Standard ML. You can view that engine as an extension of the LCF kernel, just one more rule of how to get theorems. In that instance, it is exactly the same.

[zozbot234]

How does the proof of correctness work in this case? Is the rewriting ultimately resorting to proof steps within the kernel like a LCF-tactic would, or is there simply an informal argument that whatever you're doing in the rewriter is correct, on par with the proof kernel itself?

[practal]

There is a proof as part of my thesis that the engine is correct, but it is not formal in the sense of machine-checked.

Note that the final result of the Flyspeck project does not depend on that proof, as the linear inequalities part has later on been redone and extended in HOL-Light by Alexey Solovyev, using just the LCF kernel of HOL-Light. Which proves that using a simple LCF kernel can definitely be fast enough for such computations, even on that scale!