 
Summary: Subtyping with Power Types
David Aspinall
http://www.dcs.ed.ac.uk/home/da
LFCS, University of Edinburgh, U.K.
Abstract. This paper introduces a typed calculus called Power , a
predicative reformulation of part of Cardelli's power type system. Power
types integrate subtyping into the typing judgement, allowing bounded
abstraction and bounded quantification over both types and terms. This
gives a powerful and concise system of dependent types, but leads to
difficulty in the metatheory and semantics which has impeded the ap
plication of power types so far. Basic properties of Power are proved here,
and it is given a model definition using a form of applicative structures. A
particular novelty is the auxiliary system for rough typing, which assigns
simple types to terms in Power . These "rough" types are used to prove
strong normalization of the calculus and to structure models, allowing a
novel form of containment semantics without a universal domain.
Keywords: type theory, subtyping, dependent types.
1 Introducing Power Types
Power types were introduced in a seminal paper by Cardelli [4]. The notion is
that Power (A) is a type "whose elements are all of the subtypes of the type A,"
