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

Category Theory in Coq 8.5

We report on our experience implementing category theory in Coq 8.5. The repository of this development can be found at https://bitbucket.org/amintimany/categories/. This implementation most notably makes use of features, primitive projections for records and universe polymorphism that are new to Coq 8.5.Comment: This is the abstract for a talk accepted for a presentation at the 7th Coq Workshop, Sophia Antipolis, France on June 26, 201

Formal categorical reasoning

In this paper, we present a category theory library developed in the proof assistant Coq. We discuss the design principles of the library in comparison with those existing out there. To explicitly demonstrate the utility of the library, we conclude with a case study in which a Coq formalized soundness proof of the intuitionistic propositional logic within a category theoretical settings is examined

Event-B in the Institutional Framework: Defining a Semantics, Modularisation Constructs and Interoperability for a Specification Language

Event-B is an industrial-strength specification language for verifying the properties of a given system’s specification. It is supported by its Eclipse-based IDE, Rodin, and uses the process of refinement to model systems at different levels of abstraction. Although a mature formalism, Event-B has a number of limitations. In this thesis, we demonstrate that Event-B lacks formally defined modularisation constructs. Additionally, interoperability between Event-B and other formalisms has been achieved in an ad hoc manner. Moreover, although a formal language, Event-B does not have a formal semantics. We address each of these limitations in this thesis using the theory of institutions. The theory of institutions provides a category-theoretic way of representing a formalism. Formalisms that have been represented as institutions gain access to an array of generic specification-building operators that can be used to modularise specifications in a formalismindependent manner. In the theory of institutions, there are constructs (known as institution (co)morphisms) that provide us with the facility to create interoperability between formalisms in a mathematically sound way. The main contribution of this thesis is the definition of an institution for Event-B, EVT, which allows us to address its identified limitations. To this end, we formally define a translational semantics from Event- B to EVT. We show how specification-building operators can provide a unified set of modularisation constructs for Event-B. In fact, the institutional framework that we have incorporated Event-B into is more accommodating to modularisation than the current state-of-the-art for Rodin. Furthermore, we present institution morphisms that facilitate interoperability between the respective institutions for Event-B and UML. This approach is more generic than the current approach to interoperability for Event-B and in fact, allows access to any formalism or logic that has already been defined as an institution. Finally, by defining EVT, we have outlined the steps required in order to include similar formalisms into the institutional framework. Hence, this thesis acts as a template for defining an institution for a specification language