Coq Library for First-Order Logic

This library collects several results about concerning first-order logic (FOL) that have been mechanised over the span of several projects. To get started, follow the installation instructions below and import one of the summary files in the theories folder, for instance the file FullSyntax.v providing all necessary definitions regarding syntax, natural deduction systems, and semantics. There are also summary files for specific theories, for instance Arithmetics.v providing the usual systems like PA and Q. Don't hesitate to contact the maintainers if you need help!

Note that the core definitions and results of FOL are included in the Coq library of undecidability proofs, specifically the theories/FOL folder and its subfolders on Syntax, Deduction, and Semantics. With this distribution, all content regarding undecidability is natively placed in the undecidability library.

The FOL library currently extends this core with the following content:

• Deduction: More deduction systems not included in the core library.
• Semantics: More semantics not included in the core library.
• Completeness: Completeness results for Tarski, Kripke, and algebraic semantics, constructive where possible.
• Incompleteness: An abstract and synthetic version of the first incompleteness theorem, instantiated to Robinson's Q.
• Tennenbaum: Tennenbaum's theorem stating that the natural numbers are the only computable model of PA, constructivised.
• ArithmeticalHierachy: Semantic and syntactic characterisations of the arithmetical hierarchy and an equivalence proof.
• Proofmode: A tool easing derivations in a deduction system, including a HOAS input language hiding de Bruijn encoded syntax.
• Reification: A tactic automating representability proofs of Coq predicates as first-order formulas.
• Utils: A collection of additional results needed in various projects.

Installation

We are currently planning to release this library as an OPAM package. Until then, the library can be installed manually.

First, we recommend that you create a new OPAM switch, although this is optional:

opam switch create coq-library-fol-8-16 --packages=ocaml-variants.4.14.0+options,ocaml-option-flambda
opam switch link coq-library-fol-8-16 .
eval $(opam env)

Then, install the library using:

opam repo add coq-released https://coq.inria.fr/opam/released
opam update
opam install .

Contributing

In order to compile the library for development, use the above manual installation instructions (preferrably creating a new switch), but replace the last command with

opam install . --deps-only

Compile the library using make or using opam build.

To contribute, fork the project on GitHub, add a new subdirectory for your project and your files, then file a pull request.

Contributors

• Andrej Dudenhefner
• Yannick Forster
• Marc Hermes
• Johannes Hostert
• Dominik Kirst
• Mark Koch
• Dominique Larchey-Wendling
• Niklas Mück
• Benjamin Peters
• Gert Smolka
• Dominik Wehr

Publications

Conference and Journal Papers

• On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem. Yannick Forster, Dominik Kirst, Gert Smolka. CPP'19.
• Completeness Theorems for First-Order Logic Analysed in Constructive Type Theory. Yannick Forster, Dominik Kirst, Dominik Wehr. LFCS'20
• Trakhtenbrot's Theorem in Coq - A Constructive Approach to Finite Model Theory. Dominik Kirst and Dominique Larchey-Wendling. IJCAR'20.
• Completeness Theorems for First-Order Logic Analysed in Constructive Type Theory (Extended Version). Yannick Forster, Dominik Kirst, Dominik Wehr. JLC'21.
• Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq. Dominik Kirst, Marc Hermes. ITP'21.
• Trakhtenbrot's Theorem in Coq: Finite Model Theory through the Constructive Lens. Dominik Kirst, Dominique Larchey-Wendling. LMCS'22.
• Undecidability of Dyadic First-Order Logic in Coq. Johannes Hostert, Andrej Dudenhefner, Dominik Kirst. ITP'22.
• An Analysis of Tennenbaum's Theorem in Constructive Type Theory. Marc Hermes, Dominik Kirst. FSCD'22.
• Gödel's Theorem Without Tears: Essential Incompleteness in Synthetic Computability. Dominik Kirst, Benjamin Peters. CSL'23.
• Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq. Dominik Kirst, Marc Hermes. JAR'23.

Workshop Abstracts

• A Toolbox for Mechanised First-Order Logic. Johannes Hostert, Mark Koch, Dominik Kirst. The Coq Workshop, 2021.
• Synthetic Versions of the Kleene-Post and Post's Theorem. Dominik Kirst, Niklas Mück, Yannick Forster. TYPES, 2022.
• Strong, Synthetic, and Computational Proofs of Gödel's First Incompleteness Theorem. Benjamin Peters, Dominik Kirst. TYPES, 2022.
• A Coq Library for Mechanised First-Order Logic. Dominik Kirst, Johannes Hostert, Andrej Dudenhefner, Yannick Forster, Marc Hermes, Mark Koch, Dominique Larchey-Wendling, Niklas Mück, Benjamin Peters, Gert Smolka, Dominik Wehr. The Coq Workshop, 2022.