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5

It seems to me there isn't an agreement about what "HOL" means. The OP indicates in their question that they are thinking of the formalization of higher-order logic within the formalism of the simply-typed $\lambda$-calculus. To give the question some clarity, allow me to spell out my understanding of what such a formalism comprises. Recall the ...


6

The structure you want is due to Andy Pitts, and is called a tripos. It extends the notion of hyperdoctrine, which gives a categorical model for first-order logic, with enough structure to model higher-order logic. However, much more of the work in this part of categorical logic focuses on toposes instead of triposes, which are just a little bit more ...


8

The answer is no. An old theorem of Statman states that $\beta$-equivalence in the simply-typed $\lambda$-calculus is not elementary recursive, that is, no algorithm whose running time is bounded by $2^{\vdots^{2^{|S|+|T|}}}$ for a tower of exponentials of fixed height may decide whether two simply-typed terms $S$ and $T$ are $\beta$-equivalent. The ...


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