The MetaCoq Project

Sozeau, Matthieu; Anand, Abhishek; Boulier, Simon; Cohen, Cyril; Forster, Yannick; Kunze, Fabian; Malecha, Gregory; Tabareau, Nicolás; Winterhalter, Théo

doi:10.1007/s10817-019-09540-0

Cited by 49 publications

(38 citation statements)

References 37 publications

(29 reference statements)

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“…This is the reason why we need meta-programming tools as we saw in the previous paragraph. Many different tools are available now (see § 5), and we used two of them: Ltac [8] and MetaCoq [17], which offer different advantages.…”

Section: Figure 1: Difference Between Certified and Certifying Transf...mentioning

confidence: 99%

General Automation in Coq through Modular Transformations

Blot

Prisque

Keller

et al. 2021

Electron. Proc. Theor. Comput. Sci.

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Whereas proof assistants based on Higher-Order Logic benefit from external solvers' automation, those based on Type Theory resist automation and thus require more expertise. Indeed, the latter use a more expressive logic which is further away from first-order logic, the logic of most automatic theorem provers. In this article, we develop a methodology to transform a subset of Coq goals into first-order statements that can be automatically discharged by automatic provers. The general idea is to write modular, pairwise independent transformations and combine them. Each of these eliminates a specific aspect of Coq logic towards first-order logic. As a proof of concept, we apply this methodology to a set of simple but crucial transformations which extend the local context with proven first-order assertions that make Coq definitions and algebraic types explicit. They allow users of Coq to solve non-trivial goals automatically. This methodology paves the way towards the definition and combination of more complex transformations, making Coq more accessible. * This work is funded by a Nomadic Labs-Inria collaboration.

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Section: Figure 1: Difference Between Certified and Certifying Transf...mentioning

confidence: 99%

General Automation in Coq through Modular Transformations

Blot

Prisque

Keller

et al. 2021

Electron. Proc. Theor. Comput. Sci.

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“…The only difference lies in the definition of the conversion rule, which depends on the rewrite rules in the signature. We refer the reader to [Sozeau et al 2020] for an explanation of the typing rules.…”

Section: Typing and Signature Validitymentioning

confidence: 99%

“…⋉ ⟨elim Cor | ⋄ ?P, ⋄ ?P Over , ⋄ ?P Next , ⋄ Over, •⟩ P Over , (P : Cor → □, P Over : P Over, P Next : 6 METATHEORY OF RTT In this section, we present our approach to modular confluence of RTT. The correctness of our results has been formalized in the Coq proof assistant, using the MetaCoq project [Sozeau et al 2020]. The MetaCoq project provides a solid basis for our work as confluence and subject reduction have already been formalized for PCUIC .…”

Section: Typing and Signature Validitymentioning

confidence: 99%

“…An implementation of RTT with the check of the triangle property is already available in the Agda proof assistant. Furthermore, the correctness of our approach has been formalized 1 in the Coq proof assistant, using the MetaCoq project [Sozeau et al 2020], which is a necessary step to get a trusted extension of a proof assistant. In all the lemmas that have been formalized, we refer to the Coq formalization using [file.v/lemma] to indicate the file in which the lemma has been proven and the exact name of the lemma.…”

Section: Introductionmentioning

confidence: 99%

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The taming of the rew: a type theory with computational assumptions

Cockx

Tabareau

Winterhalter

2021

Proc. ACM Program. Lang.

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Dependently typed programming languages and proof assistants such as Agda and Coq rely on computation to automatically simplify expressions during type checking. To overcome the lack of certain programming primitives or logical principles in those systems, it is common to appeal to axioms to postulate their existence. However, one can only postulate the bare existence of an axiom, not its computational behaviour. Instead, users are forced to postulate equality proofs and appeal to them explicitly to simplify expressions, making axioms dramatically more complicated to work with than built-in primitives. On the other hand, the equality reflection rule from extensional type theory solves these problems by collapsing computation and equality, at the cost of having no practical type checking algorithm. This paper introduces Rewriting Type Theory (RTT), a type theory where it is possible to add computational assumptions in the form of rewrite rules. Rewrite rules go beyond the computational capabilities of intensional type theory, but in contrast to extensional type theory, they are applied automatically so type checking does not require input from the user. To ensure type soundness of RTT—as well as effective type checking—we provide a framework where confluence of user-defined rewrite rules can be checked modularly and automatically, and where adding new rewrite rules is guaranteed to preserve subject reduction. The properties of RTT have been formally verified using the MetaCoq framework and an implementation of rewrite rules is already available in the Agda proof assistant.

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“…With all of that in place, we can use extraction to obtain an OCaml implementation that satisfies the given specifications [10]. Note that the extraction mechanism of Coq was proven to be correct using MetaCoq [12].…”

Section: Introductionmentioning

confidence: 99%

Formalizing Higher-Order Termination in Coq

Vale¹,

Weide²

2021

Preprint

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We describe a formalization of higher-order rewriting theory and formally prove that an AFS is strongly normalizing if it can be interpreted in a well-founded domain. To do so, we use Coq, which is a proof assistant based on dependent type theory. Using this formalization, one can implement several termination techniques, like the interpretation method or dependency pairs, and prove their correctness. Those implementations can then be extracted to OCaml, which results in a verified termination checker.

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The MetaCoq Project

Cited by 49 publications

References 37 publications

General Automation in Coq through Modular Transformations

General Automation in Coq through Modular Transformations

The taming of the rew: a type theory with computational assumptions

Formalizing Higher-Order Termination in Coq

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