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$\begingroup$ I think I should back up. I have a distinct memory of a developer of CoQ telling me that the kernel of CoQ was written in a strongly renormalizing rewrite system EXACTLY BECAUSE it was proven to halt. That this kernel is at the base of all other systems and that the developers took pains to make sure that it is walled off so that nothing but this strongly normalizing rewrite system was allowed. The upshot was that, "On the basis that we haven't made any bugs in the kernel, therefore CoQ can be shown to halt." or something to that effect. $\endgroup$
– user3074
Commented Dec 6, 2023 at 13:50
$\begingroup$ This is I believe where I understood this from: lix.polytechnique.fr/Labo/Bruno.Barras/publi/coqincoq.pdf $\endgroup$
– user3074
Commented Dec 6, 2023 at 18:50
$\begingroup$ Yes, Coq does pin some aspects of its soundness proof on normalization, but Coq is also a complex theory and proofs in the literature don't necessarily apply to everything Coq can do in practice. This is not the case for all systems, and in particular Lean does not depend on strong normalization for its soundness argument. Also note: "strongly normalizing rewrite system EXACTLY BECAUSE it was proven to halt." - this is tautological, if it is strongly normalizing then reduction halts by definition. But maybe you should consider why you think halting is important? $\endgroup$
– Mario Carneiro
Commented Dec 7, 2023 at 1:29
$\begingroup$ Yes, I think I may have placed too much import on the fact that constructive logical systems - when embodied by computational systems always halt because they are type checked. For me it has always seemed a hugely important fact that you can't build a non-halting program in a constructive system. If I tried to explicitly program a non-halting program in the simple typed lambda calculus or any of the systems in the lambda cube I couldn't do it because they are all strongly normalizing. $\endgroup$
– user3074
Commented Dec 7, 2023 at 3:11

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