Type theoretical databases

Henrik Forssell, Håkon Robbestad Gylterud, David I. Spivak

https://arxiv.org/abs/1406.6268

It seems that for the current results it would be enough to consider a theory with a single univalent universe of sets (aka the Hofmann-Streicher groupoid model, which had been known before “HoTT” came up). Is that right? I am curious whether it is possible to get a significant advantage from unrestricted types. I hope we can read about it here when you find out 🙂

I found it indeed a bit surprising that you formalise a schema as a tree. I guess I would have tried vectors first (maybe implemented as maps from some finite type Fin n to the universe). Any (order-preserving) projection to a subset of the components would then just be given as a composition with an injective (and order-preserving) map between finite sets. This would probably make some proofs a bit harder but it would avoid the necessity of nesting ‘lefts’ and ‘rights’ (the nesting seems a bit arbitrary to me).

]]>A small comment: The type that you discuss in the paragraph “Constructive Finiteness”, i.e. the type of types that are equivalent to some , is equivalent to the natural numbers. This is because the first component (a type ) and the third component (an equality ) form a “singleton pair” and cancel each other out; see the HoTT book Lemma 3.11.8. ]]>

I agree with you re: “why finite sets” and combinatorics. Perhaps a better way to ask the question I am really asking is: what, if anything, do we *lose* if we drop the “finite” restriction? Or put conversely, where does the theory of species make use of finiteness in an essential way? This is one question a proof assistant formalization would really help answer.

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