J Autom Reasoning
From Types to Sets by Local Type Deﬁnition in
· Andrei Popescu
Received: 31 March 2017 / Accepted: 20 April 2018
© Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract Types in higher-order logic (HOL) are naturally interpreted as nonempty sets.
This intuition is reﬂected in the type deﬁnition rule for the HOL-based systems (including
Isabelle/HOL), where a new type can be deﬁned whenever a nonempty set is exhibited.
However, in HOL this deﬁnition mechanism cannot be applied inside proof contexts.We
propose a more expressive type deﬁnition rule that addresses the limitation and we prove
its consistency. This higher expressive power opens the opportunity for a HOL tool that
relativizes type-based statements to more ﬂexible set-based variants in a principled way. We
also address particularities of Isabelle/HOL and show how to perform the relativization in
the presence of type classes.
Keywords HOL · Isabelle · Local typedef · Type deﬁnition · Relativization · Type classes ·
Overloading · Dependent types · Model · Consistency · Transfer · Type-based theorems ·
The proof assistant community is mainly divided in two successful camps. One camp, rep-
resented by provers such as Agda , Coq , Matita  and Nuprl , uses expressive
type theories as a foundation. The other camp, represented by the HOL family of provers
This is the extended, journal version of the conference paper , submitted to the JAR special issue
dedicated to ITP 2016.
Fakultät für Informatik, Technische Universität München, Munich, Germany
Department of Computer Science, School of Science and Technology, Middlesex University, London,