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DOI: 10.4230/LIPIcs.ITP.2024.26

The Rewster: Type Preserving Rewrite Rules for the Coq Proof Assistant

Authors: Yann Leray, Gaëtan Gilbert, Nicolas Tabareau, and Théo Winterhalter

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract

In dependently typed proof assistants, users can declare axioms to extend the ambient logic locally with new principles and propositional equalities governing them. Additionally, rewrite rules have recently been proposed to allow users to extend the logic with new definitional equalities, enabling them to handle new principles with a computational behaviour. While axioms can only break consistency, the addition of arbitrary rewrite rules can break other important metatheoretical properties such as type preservation. In this paper, we present an implementation of rewrite rules on top of the Coq proof assistant, together with a modular criterion to ensure that the added rewrite rules preserve typing. This criterion, based on bidirectional type checking, is formally expressed in PCUIC - the type theory of Coq recently developed in the MetaCoq project.

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Yann Leray, Gaëtan Gilbert, Nicolas Tabareau, and Théo Winterhalter. The Rewster: Type Preserving Rewrite Rules for the Coq Proof Assistant. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{leray_et_al:LIPIcs.ITP.2024.26,
  author =	{Leray, Yann and Gilbert, Ga\"{e}tan and Tabareau, Nicolas and Winterhalter, Th\'{e}o},
  title =	{{The Rewster: Type Preserving Rewrite Rules for the Coq Proof Assistant}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{26:1--26:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.26},
  URN =		{urn:nbn:de:0030-drops-207545},
  doi =		{10.4230/LIPIcs.ITP.2024.26},
  annote =	{Keywords: type theory, dependent types, rewrite rules, type preservation, Coq}
}

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DOI: 10.4230/LIPIcs.TLCA.2015.226

Wild omega-Categories for the Homotopy Hypothesis in Type Theory

Authors: André Hirschowitz, Tom Hirschowitz, and Nicolas Tabareau

Published in: LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

Abstract

In classical homotopy theory, the homotopy hypothesis asserts that the fundamental varpi-groupoid construction induces an equivalence between topological spaces and weak varpi-groupoids. In the light of Voevodsky's univalent foundations program, which puts forward an interpretation of types as topological spaces, we consider the question of transposing the homotopy hypothesis to type theory. Indeed such a transposition could stand as a new approach to specifying higher inductive types. Since the formalisation of general weak varpi-groupoids in type theory is a difficult task, we only take a first step towards this goal, which consists in exploring a shortcut through strict varpi-categories. The first outcome is a satisfactory type-theoretic notion of strict varpi-category, which has hsets of cells in all dimensions. For this notion, defining the 'fundamental strict varpi-category' of a type seems out of reach. The second outcome is an 'incoherently strict' notion of type-theoretic varpi-category, which admits arbitrary types of cells in all dimensions. These are the 'wild' varpi-categories of the title. They allow the definition of a 'fundamental wild varpi-category' map, which leads to our (partial) homotopy hypothesis for type theory (stating an adjunction, not an equivalence). All of our results have been formalised in the Coq proof assistant. Our formalisation makes systematic use of the machinery of coinductive types.

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André Hirschowitz, Tom Hirschowitz, and Nicolas Tabareau. Wild omega-Categories for the Homotopy Hypothesis in Type Theory. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 226-240, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hirschowitz_et_al:LIPIcs.TLCA.2015.226,
  author =	{Hirschowitz, Andr\'{e} and Hirschowitz, Tom and Tabareau, Nicolas},
  title =	{{Wild omega-Categories for the Homotopy Hypothesis in Type Theory}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{226--240},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-87-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{38},
  editor =	{Altenkirch, Thorsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.226},
  URN =		{urn:nbn:de:0030-drops-51669},
  doi =		{10.4230/LIPIcs.TLCA.2015.226},
  annote =	{Keywords: Homotopy Type theory; Omega-categories; Coinduction; Homotopy hypothesis}
}

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Documents authored by Tabareau, Nicolas

The Rewster: Type Preserving Rewrite Rules for the Coq Proof Assistant

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Wild omega-Categories for the Homotopy Hypothesis in Type Theory

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