This paper introduces an expressive class of indexed quotient-inductive types, called QWI within the framework constructive type theory. They are initial algebras for families equational theories with possibly infinitary operators and equations. We prove that types can be derived from quotient inductive in theory toposes natural number object universes, provided those universes satisfy Weakly I...

We describe the categorical semantics for a simply typed variant and simplified dependently of Cocon , contextual modal type theory where box modality mediates between weak function space that is used to represent higher-order abstract syntax (HOAS) trees strong describes (recursive) computations about them. What makes different from standard theories presence first-class contexts objects are c...

A co-type is a type with instance methods and instance data which enhance the functionality of some other type (its "expanded" type). The instance methods of the co-type correspond approximately to constructors, class methods and binary methods of the expanded type in other class-based OO systems, and the instance data replaces class-based data. This unconventional approach was motivated by the...

Types are an important part of any modern programming language, but we often forget that the concept of type we understand nowadays is not the same it was perceived in the sixties. Moreover, we conflate the concept of “type” in programming languages with the concept of the same name in mathematical logic, an identification that is only the result of the convergence of two different paths, which...

We present two mutual encodings, respectively of the Calculus of Inductive Constructions in Zermelo-Frrnkel set theory and the opposite way. More precisely, we actually construct two families of en-codings, relating the number of universes in the type theory with the number of inaccessible cardinals in the set theory. The main result is that both hierarchies of logical formalisms interleave w.r...

We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive types exist in any Martin-Löf category (extensive locally cartesian closed category with W-types) by exploiting our work on container types. This generalises a result by Dybjer (1997) who showed that non-nested strictly positive inductive types can be re...

Data Types without the Types Philip Wadler (Edinburgh University, Scotland [email protected]) Dedicated to David Turner on the occasion of his 70’th birthday Abstract: The data abstraction mechanism of Miranda may be adapted to a dynamically typed programming language by applying ideas from gradual typing.