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DOI: 10.4230/LIPIcs.FSCD.2023.18

For the Metatheory of Type Theory, Internal Sconing Is Enough

Authors: Rafaël Bocquet, Ambrus Kaposi, and Christian Sattler

Published in: LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)

Abstract

Metatheorems about type theories are often proven by interpreting the syntax into models constructed using categorical gluing. We propose to use only sconing (gluing along a global section functor) instead of general gluing. The sconing is performed internally to a presheaf category, and we recover the original glued model by externalization. Our method relies on constructions involving two notions of models: first-order models (with explicit contexts) and higher-order models (without explicit contexts). Sconing turns a displayed higher-order model into a displayed first-order model. Using these, we derive specialized induction principles for the syntax of type theory. The input of such an induction principle is a boilerplate-free description of its motives and methods, not mentioning contexts. The output is a section with computation rules specified in the same internal language. We illustrate our framework by proofs of canonicity and normalization for type theory.

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Rafaël Bocquet, Ambrus Kaposi, and Christian Sattler. For the Metatheory of Type Theory, Internal Sconing Is Enough. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 18:1-18:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bocquet_et_al:LIPIcs.FSCD.2023.18,
  author =	{Bocquet, Rafa\"{e}l and Kaposi, Ambrus and Sattler, Christian},
  title =	{{For the Metatheory of Type Theory, Internal Sconing Is Enough}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{18:1--18:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-277-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{260},
  editor =	{Gaboardi, Marco and van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.18},
  URN =		{urn:nbn:de:0030-drops-180029},
  doi =		{10.4230/LIPIcs.FSCD.2023.18},
  annote =	{Keywords: type theory, presheaves, canonicity, normalization, sconing, gluing}
}

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DOI: 10.4230/LIPIcs.TYPES.2021.3

Strictification of Weakly Stable Type-Theoretic Structures Using Generic Contexts

Authors: Rafaël Bocquet

Published in: LIPIcs, Volume 239, 27th International Conference on Types for Proofs and Programs (TYPES 2021)

Abstract

We present a new strictification method for type-theoretic structures that are only weakly stable under substitution. Given weakly stable structures over some model of type theory, we construct equivalent strictly stable structures by evaluating the weakly stable structures at generic contexts. These generic contexts are specified using the categorical notion of familial representability. This generalizes the local universes method of Lumsdaine and Warren. We show that generic contexts can also be constructed in any category with families which is freely generated by collections of types and terms, without any definitional equality. This relies on the fact that they support first-order unification. These free models can only be equipped with weak type-theoretic structures, whose computation rules are given by typal equalities. Our main result is that any model of type theory with weakly stable weak type-theoretic structures admits an equivalent model with strictly stable weak type-theoretic structures.

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Rafaël Bocquet. Strictification of Weakly Stable Type-Theoretic Structures Using Generic Contexts. In 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 239, pp. 3:1-3:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bocquet:LIPIcs.TYPES.2021.3,
  author =	{Bocquet, Rafa\"{e}l},
  title =	{{Strictification of Weakly Stable Type-Theoretic Structures Using Generic Contexts}},
  booktitle =	{27th International Conference on Types for Proofs and Programs (TYPES 2021)},
  pages =	{3:1--3:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-254-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{239},
  editor =	{Basold, Henning and Cockx, Jesper and Ghilezan, Silvia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2021.3},
  URN =		{urn:nbn:de:0030-drops-167724},
  doi =		{10.4230/LIPIcs.TYPES.2021.3},
  annote =	{Keywords: type theory, strictification, coherence, familial representability, unification}
}

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Documents authored by Bocquet, Rafaël

For the Metatheory of Type Theory, Internal Sconing Is Enough

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Strictification of Weakly Stable Type-Theoretic Structures Using Generic Contexts

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