Bertus: Implementing Observational Equality

The marriage between programming and logic has been a fertile one. In particular, since the definition of the simply typed λ-calculus, a number of type systems have been devised with increasing expressive power. Among this systems, Intuitionistic Type Theory (ITT) has been a popular framework for theorem provers and programming languages. However, reasoning about equality has always been a tricky business in ITT and related theories. In this thesis we shall explain why this is the case, and present Observational Type Theory (OTT), a solution to some of the problems with equality. To bring OTT closer to the current practice of interactive theorem provers, we describe Bertus, a system featuring OTT in a setting more close to the one found in widely used provers such as Agda and Coq. Most notably, we feature user defined inductive and record types and a cumulative, implicit type hierarchy. Having implemented part of Bertus as a Haskell program, we describe some of the implementation issues faced. Acknowledgements I would like to thank Steffen van Bakel, my supervisor, who was brave enough to believe in my project and who provided support and invaluable advice. I would also like to thank the Haskell and Agda community on IRC, which guided me through the strange world of types; and in particular Andrea Vezzosi and James Deikun, with whom I entertained countless insightful discussions over the past year. Andrea suggested Observational Type Theory as a topic of study: this thesis would not exist without him. Before them, Tony Field introduced me to Haskell, unknowingly filling most of my free time from that time on. Finally, most of the work stems from the research of Conor McBride, who answered many of my doubts through these months. I also owe him the colours.

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