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Build status MetaCoq Chat Open in Visual Studio Code

MetaCoq is a project formalizing Coq in Coq and providing tools for manipulating Coq terms and developing certified plugins (i.e. translations, compilers or tactics) in Coq.

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Getting started

Installation instructions




Overview of the project

At the center of this project is the Template-Coq quoting library for Coq. The project currently has a single repository extending Template-Coq with additional features. Each extension is in a dedicated folder. The dependency graph might be useful to navigate the project. Statistics: ~300kLoC of Coq, ~30kLoC of OCaml.

Template-Coq is a quoting library for Coq. It takes Coq terms and constructs a representation of their syntax tree as an inductive data type. The representation is based on the kernel's term representation.

After importing MetaCoq.Template.Loader there are commands MetaCoq Test Quote t., MetaCoq Quote Definition name := (t). and MetaCoq Quote Recursively Definition name := (t). as well as a tactic quote_term t k, where in all cases t is a term and k a continuation tactic.

In addition to this representation of terms, Template Coq includes:

  • Reification of the environment structures, for constant and inductive declarations along with their universe structures.

  • Denotation of terms and global declarations.

  • A monad for querying the environment, manipulating global declarations, calling the type checker, and inserting them in the global environment, in the style of MTac. Monadic programs p : TemplateMonad A can be run using MetaCoq Run p.

  • A formalization of the typing rules reflecting the ones of Coq, covering all of Coq except eta-expansion and template polymorphism.

PCUIC, the Polymorphic Cumulative Calculus of Inductive Constructions is a cleaned up version of the term language of Coq and its associated type system, shown equivalent to the one of Coq. This version of the calculus has proofs of standard metatheoretical results:

  • Weakening for global declarations, weakening and substitution for local contexts.

  • Confluence of reduction using a notion of parallel reduction

  • Context cumulativity / conversion and validity of typing.

  • Subject Reduction (case/cofix reduction excluded)

  • Principality: every typeable term has a smallest type.

  • Bidirectional presentation: an equivalent presentation of the system that enforces directionality of the typing rules. Strengthening follows from this presentation.

  • Elimination restrictions: the elimination restrictions ensure that singleton elimination (from Prop to Type) is only allowed on singleton inductives in Prop.

  • Canonicity: The weak head normal form of a term of inductive type is a constructor application.

  • Consistency under the assumption of strong normalization

  • Weak call-by-value standardization: Normal forms of terms of first-order inductive type can be found via weak call-by-value evaluation.

See the PCUIC README for a detailed view of the development.

Implementation of a fuel-free and verified reduction machine, conversion checker and type checker for PCUIC. This relies on a postulate of strong normalization of the reduction relation of PCUIC on well-typed terms. The checker is shown to be correct and complete w.r.t. the PCUIC specification. The extracted safe checker is available in Coq through a new vernacular command:

MetaCoq SafeCheck <term>

After importing MetaCoq.SafeChecker.Loader.

To roughly compare the time used to check a definition with Coq's vanilla type-checker, one can use:

MetaCoq CoqCheck <term>

This also includes a verified, efficient re-typing procedure (useful in tactics) in MetaCoq.SafeChecker.PCUICSafeRetyping.

See the SafeChecker README for a detailed view of the development.

An erasure procedure to untyped lambda-calculus accomplishing the same as the type and proof erasure phase of the Extraction plugin of Coq. The extracted safe erasure is available in Coq through a new vernacular command:

MetaCoq Erase <term>

After importing MetaCoq.Erasure.Loader.

The erasure pipeline includes verified optimizations to remove lets in constructors, remove cases on propositional terms, switch to an unguarded fixpoint reduction rule and transform the higher-order constructor applications to first-order blocks for easier translation to usual programming languages. See the erasure README for a detailed view of the development.

Examples of translations built on top of this:

The Quotation module is geared at providing functions □T → □□T for □T := Ast.term (currently implemented) and for □T := { t : Ast.term & Σ ;;; [] |- t : T } (still in the works).

Ultimately the goal of this development is to prove that is a lax monoidal semicomonad (a functor with cojoin : □T → □□T that codistributes over unit and ×), which is sufficient for proving Löb's theorem.

The public-facing interface of this development is provided in MetaCoq.Quotation.ToTemplate.All and MetaCoq.Quotation.ToPCUIC.All.

See the Quotation README for a more detailed view of the development.



Related Projects

  • The CertiCoq project develops a certified compiler from the output of verified erasure down to CompCert C-light. It provides in particular OCaml and fully foundationally verified plugins for the whole compilation pipeline from Gallina to Clight and the verified type-checker of MetaCoq.

  • The ConCert project develops certified or certifying compilers from Gallina to smart contract languages (Liquidity and CameLIGO), the functional language Elm, and a subset of the Rust programming languages. It uses the typed erasure variant to gather more type information about erased terms and perform optimizations based on this information. The project focuses in particular on the verification and safe extraction of smart contracts for blockchains.

Team & Credits

Abhishek Anand Danil Annenkov Simon Boulier
Cyril Cohen Yannick Forster Jason Gross
Meven Lennon-Bertrand Kenji Maillard Gregory Malecha
Jakob Botsch Nielsen Matthieu Sozeau Nicolas Tabareau
Théo Winterhalter

MetaCoq is developed by (left to right) Abhishek Anand, Danil Annenkov, Simon Boulier, Cyril Cohen, Yannick Forster, Jason Gross, Meven Lennon-Bertrand, Kenji Maillard, Gregory Malecha, Jakob Botsch Nielsen, Matthieu Sozeau, Nicolas Tabareau and Théo Winterhalter.

Copyright (c) 2014-2023 Gregory Malecha
Copyright (c) 2015-2023 Abhishek Anand, Matthieu Sozeau
Copyright (c) 2017-2023 Simon Boulier, Nicolas Tabareau, Cyril Cohen
Copyright (c) 2018-2023 Danil Annenkov, Yannick Forster, Théo Winterhalter
Copyright (c) 2020-2023 Jakob Botsch Nielsen, Meven Lennon-Bertrand
Copyright (c) 2022-2023 Kenji Maillard
Copyright (c) 2023      Jason Gross

This software is distributed under the terms of the MIT license. See LICENSE for details.


Please report any bugs or feature requests on the github issue tracker.