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Shallow Embedding of Type Theory is Morally Correct

Kaposi, Ambrus; Kov�cs, Andr�s; Kraus, Nicolai

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Authors

Ambrus Kaposi

Andr�s Kov�cs



Abstract

© 2019, Springer Nature Switzerland AG. There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary to refer to the syntax, for example in proofs of canonicity and normalisation. One option is to embed the syntax deeply, by using inductive definitions in a proof assistant. However, in this case the handling of definitional equalities becomes technically challenging. Alternatively, we can reuse conversion checking in the metatheory by shallowly embedding the object theory. In this paper, we consider the standard model of a type theoretic object theory in Agda. This model has the property that all of its equalities hold definitionally, and we can use it as a shallow embedding by building expressions from the components of this model. However, if we are to reason soundly about the syntax with this setup, we must ensure that distinguishable syntactic constructs do not become provably equal when shallowly embedded. First, we prove that shallow embedding is injective upto definitional equality, by modelling the embedding as a syntactic translation targeting the metatheory. Second, we use an implementation hiding trick to disallow illegal propositional equality proofs and constructions which do not come from the syntax. We showcase our technique with very short formalisations of canonicity and parametricity for Martin-Löf type theory. Our technique only requires features which are available in all major proof assistants based on dependent type theory.

Citation

Kaposi, A., Kovács, A., & Kraus, N. (2019). Shallow Embedding of Type Theory is Morally Correct. Lecture Notes in Artificial Intelligence, 11825 LNCS, 329-365. https://doi.org/10.1007/978-3-030-33636-3_12

Journal Article Type Conference Paper
Conference Name International Conference on Mathematics of Program Construction
Conference Location Porto, Portugal
Acceptance Date Aug 22, 2019
Online Publication Date Oct 20, 2019
Publication Date Oct 20, 2019
Deposit Date Jul 15, 2020
Publicly Available Date Mar 28, 2024
Journal Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Electronic ISSN 1611-3349
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 11825 LNCS
Pages 329-365
Series Title Lecture Notes in Computer Science
Series Number 11825
DOI https://doi.org/10.1007/978-3-030-33636-3_12
Public URL https://nottingham-repository.worktribe.com/output/4373315
Publisher URL https://link.springer.com/book/10.1007/978-3-030-33636-3

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