Dependent types reflect the fact that validity of data is often a relative notion by allowing prior data to affect the types of subsequent data. Not only does this make for a precise type system, but also a highly generic one: both the type and the program for each instance of a family of operations can be computed from the data which codes for that instance. Recent experimental extensions to the Haskell type class mechanism give us strong tools to relativize types to other types. We may simulate some aspects of dependent typing by making counterfeit type-level copies of data, with type constructors simulating data constructors and type classes simulating datatypes. This paper gives examples of the technique and discusses its potential. 1

Abstract. This paper gives a reduction-preserving translation from Coquand’s dependent pattern matching [4] into a traditional type theory [11] with universes, inductive types and relations and the axiom K [22]. This translation serves as a proof of termination for structurally recursive pattern matching programs, provides an implementable compilation technique in the style of functional programming languages, and demonstrates the equivalence with a more easily understood type theory. Dedicated to Professor Joseph Goguen on the occasion of his 65th birthday. 1

The Calculus of Constructions (CC) ([Coquand 1985]) is a typed lambda calculus for higher order intuitionistic logic: proofs of the higher order logic are interpreted as lambda terms and formulas as types. It is also the union of Girard's system F! ([Girard 1972]), a higher order typed lambda calculus, and a first order dependent typed lambda calculus in the style of de Bruijn's Automath ([de Bruijn 1980]) or Martin-Lof's intuitionistic theory of types ([Martin-Lof 1984]). Using the impredicative coding of data types in F! , the Calculus of Constructions thus becomes a higher order language for the typing of functional programs. We shall introduce and try to explain CC by exploiting especially the first point of view, by introducing a typed lambda calculus that faithfully represent higher order predicate logic (so for this system the Curry-Howard `formulas-as-types isomorphism' is really an isomorphism.) Then we discuss some propositions that are provable in CC but not in the higher or...

We construct a logic-enriched type theory LTTw that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTw, including Weyl’s definition of the cardinality of a set and several results from real analysis, using the proof assistant Plastic that implements the logical framework LF. This case study shows how type theory can be used to represent a non-constructive foundation for mathematics.