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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}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2016.5

Design and Implementation of the Andromeda Proof Assistant

Authors: Andrej Bauer, Gaëtan Gilbert, Philipp G. Haselwarter, Matija Pretnar, and Christopher A. Stone

Published in: LIPIcs, Volume 97, 22nd International Conference on Types for Proofs and Programs (TYPES 2016)

Abstract

Andromeda is an LCF-style proof assistant where the user builds derivable judgments by writing code in a meta-level programming language AML. The only trusted component of Andromeda is a minimalist nucleus (an implementation of the inference rules of an object-level type theory), which controls construction and decomposition of type-theoretic judgments. Since the nucleus does not perform complex tasks like equality checking beyond syntactic equality, this responsibility is delegated to the user, who implements one or more equality checking procedures in the meta-language. The AML interpreter requests witnesses of equality from user code using the mechanism of algebraic operations and handlers. Dynamic checks in the nucleus guarantee that no invalid object-level derivations can be constructed. To demonstrate the flexibility of this system structure, we implemented a nucleus consisting of dependent type theory with equality reflection. Equality reflection provides a very high level of expressiveness, as it allows the user to add new judgmental equalities, but it also destroys desirable meta-theoretic properties of type theory (such as decidability and strong normalization). The power of effects and handlers in AML is demonstrated by a standard library that provides default algorithms for equality checking, computation of normal forms, and implicit argument filling. Users can extend these new algorithms by providing local "hints" or by completely replacing these algorithms for particular developments. We demonstrate the resulting system by showing how to axiomatize and compute with natural numbers, by axiomatizing the untyped lambda-calculus, and by implementing a simple automated system for managing a universe of types.

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Andrej Bauer, Gaëtan Gilbert, Philipp G. Haselwarter, Matija Pretnar, and Christopher A. Stone. Design and Implementation of the Andromeda Proof Assistant. In 22nd International Conference on Types for Proofs and Programs (TYPES 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 97, pp. 5:1-5:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bauer_et_al:LIPIcs.TYPES.2016.5,
  author =	{Bauer, Andrej and Gilbert, Ga\"{e}tan and Haselwarter, Philipp G. and Pretnar, Matija and Stone, Christopher A.},
  title =	{{Design and Implementation of the Andromeda Proof Assistant}},
  booktitle =	{22nd International Conference on Types for Proofs and Programs (TYPES 2016)},
  pages =	{5:1--5:31},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-065-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{97},
  editor =	{Ghilezan, Silvia and Geuvers, Herman and Ivetic, Jelena},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2016.5},
  URN =		{urn:nbn:de:0030-drops-98574},
  doi =		{10.4230/LIPIcs.TYPES.2016.5},
  annote =	{Keywords: type theory, proof assistant, equality reflection, computational effects}
}

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Documents authored by Gilbert, Gaëtan

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

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Design and Implementation of the Andromeda Proof Assistant

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