Experience Implementing a Performant Category-Theory Library in Coq

We describe our experience implementing a broad category-theory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed substantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share the lessons we have learned about how to represent very abstract mathematical objects and arguments in Coq and how future proof assistants might be designed to better support such reasoning. One particular encoding trick to which we draw attention allows category-theoretic arguments involving duality to be internalized in Coq's logic with definitional equality. Ours may be the largest Coq development to date that uses the relatively new Coq version developed by homotopy type theorists, and we reflect on which new features were especially helpful.Comment: The final publication will be available at link.springer.com. This version includes a full bibliography which does not fit in the Springer version; other than the more complete references, this is the version submitted as a final copy to ITP 201

Point-Free, Set-Free Concrete Linear Algebra

International audienceWe show how a simple variant of Gaussian elimination can be used to model abstract linear algebra directly, using matrices only to represent all categories of objects, with operations such as subspace intersection and sum. We can even provide effective support for direct sums and subalgebras. We have formalized this work in Coq, and used it to develop all of the group representation theory required for the proof of the Odd Order Theorem, including results such as the Jacobson Density Theorem, Clifford's Theorem, the Jordan-Holder Theorem for modules, the Wedderburn Structure Theorem for semisimple rings (the basis for character theory).On présente une formalisation en Coq de l'algèbre linéaire où tous les objets sont représentés par des matrices, y compris les sous-espaces. Ce développement a été utilisé pour élaborer la formalisation des éléments de théorie de la représentation nécessaires à la prévue du théorème de Feit-Thompson

Formalizing of Category Theory in Agda

The generality and pervasiness of category theory in modern mathematics makes it a frequent and useful target of formalization. It is however quite challenging to formalize, for a variety of reasons. Agda currently (i.e. in 2020) does not have a standard, working formalization of category theory. We document our work on solving this dilemma. The formalization revealed a number of potential design choices, and we present, motivate and explain the ones we picked. In particular, we find that alternative definitions or alternative proofs from those found in standard textbooks can be advantageous, as well as "fit" Agda's type theory more smoothly. Some definitions regarded as equivalent in standard textbooks turn out to make different "universe level" assumptions, with some being more polymorphic than others. We also pay close attention to engineering issues so that the library integrates well with Agda's own standard library, as well as being compatible with as many of supported type theories in Agda as possible

Construction of real algebraic numbers in Coq

This paper shows a construction in Coq of the set of real algebraic numbers, together with a formal proof that this set has a structure of discrete archimedian real closed field. This construction hence implements an interface of real closed field. Instances of such an interface immediately enjoy quantifier elimination thanks to a previous work. This work also intends to be a basis for the construction of complex algebraic numbers and to be a reference implementation for the certification of numerous algorithms relying on algebraic numbers in computer algebra

Importing HOL Light into Coq

Abstract. We present a new scheme to translate mathematical developments from HOL Light to Coq, where they can be re-used and rechecked. By relying on a carefully chosen embedding of Higher-Order Logic into Type Theory, we try to avoid some pitfalls of inter-operation between proof systems. In particular, our translation keeps the mathematical statements intelligible. This translation has been implemented and allows the importation of the HOL Light basic library into Coq.

TraceContract: A Scala DSL for Trace Analysis

Abstract. In this paper we describe TRACECONTRACT, an API for trace analysis, implemented in the SCALA programming language. We argue that for certain forms of trace analysis the best weapon is a high level programming language augmented with constructs for temporal reasoning. A trace is a sequence of events, which may for example be generated by a running program, instrumented appropriately to generate events. The API supports writing properties in a notation that combines an advanced form of data parameterized state machines with temporal logic. The implementation utilizes SCALA’s support for defining internal Domain Specific Languages (DSLs). Furthermore SCALA’s combination of object oriented and functional programming features, including partial functions and pattern matching, makes it an ideal host language for such an API.

A Language of Patterns for Subterm Selection

International audienceThis paper describes the language of patterns that equips the SSReflect proof shell extension for the Coq system. Patterns are used to focus proof commands on sub expressions of the conjecture under analysis in a declarative manner. They are designed to ease the writing of proof scripts and to increase their readability and maintainability. A pattern can identify the sub expression of interest approximating the sub expression itself, or its enclosing context or both. The user is free to choose the most convenient option. Patterns are matched following an extremely precise and predictable discipline, that is carefully designed to admit an efficient implementation. In this paper we report on the language of patterns, its matching algorithm and its usage in the formal library developed by the Mathematical Components team to support the verification of the Odd Order theorem

Competing inheritance paths in dependent type theory: a case study in functional analysis

International audienc