Abstract
Theories are an essential structuring principle that enable modularity, encapsulation, and reuse in formal libraries and programs (called classes there). Similar effects can be achieved by dependent record types. While the former form a separate language layer, the latter are a normal part of the type theory. This overlap in functionality can render different systems non-interoperable and lead to duplication of work.
We present a type-theoretic calculus and implementation of a variant of record types that for a wide class of formal languages naturally corresponds to theories. Moreover, we can now elegantly obtain a contravariant functor that reflects the theory level into the object level: for each theory we obtain the type of its models and for every theory morphism a function between the corresponding types. In particular this allows shallow – and thus structure-preserving – encodings of mathematical knowledge and program specifications while allowing the use of object-level features on models, e.g. equality and quantification.
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- 1.
This is sometimes called horizontal subtyping. In that case, the straightforward covariance rule for record types is called vertical subtyping.
- 2.
Note that it does not matter how the fields of the record are split into \(\varDelta _1\) and \(\varDelta _2\) as long as \(\varDelta _1\) contains all fields that the declaration of x depends on.
- 3.
It is possible and important in practice to also define \(\varDelta \oplus E\) when the types/definitions in d and e are provably equal. We omit that here for simplicity.
- 4.
Note that because \(P\cap P'\) depends on the syntactic structure of P and \(P'\), it only approximates the least upper bound of \(\mathtt {Mod}\left( P\right) \) and \(\mathtt {Mod}\left( P'\right) \) and is not stable under, e.g., flattening of P and \(P'\). But it can still be very useful in certain situations.
- 5.
The computation of \(\max P\) may look like it requires flattening. But it is easy to compute and cache its value for every named theory.
- 6.
In practice, these substitutions are easy to implement without flattening r because we can cache for every theory which theories it includes and which names it declares.
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Acknowledgements
The work reported here has been kicked off by discussions with Jacques Carette and William Farmer who have experimented with theory internalizations into record types in the scope of their MathScheme system. We acknowledge financial support from the OpenDreamKit Horizon 2020 European Research Infrastructures project (#676541).
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Müller, D., Rabe, F., Kohlhase, M. (2018). Theories as Types. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds) Automated Reasoning. IJCAR 2018. Lecture Notes in Computer Science(), vol 10900. Springer, Cham. https://doi.org/10.1007/978-3-319-94205-6_38
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DOI: https://doi.org/10.1007/978-3-319-94205-6_38
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