this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper [24] describes the basic features of the LF. In this paper we are going to provide a broader illustration of its applicability and discuss to what extent it is successful. The analysis (of the formal presentation) of a system carried out through encoding often illuminates the system itself. This paper will also deal with this phenomenon.

Quest is a programming language based on impredicative type quantifiers and subtyping within a three-level structure of kinds, types and type operators, and values. The semantics of Quest is rather challenging. In particular, difficulties arise when we try to model simultaneously features such as contravariant function spaces, record types, subtyping, recursive types, and fixpoints. In this paper we describe in detail the type inference rules for Quest, and we give them meaning using a partial equivalence relation model of types. Subtyping is interpreted as in previous work by Bruce and Longo, but the interpretation of some aspects, namely subsumption, power kinds, and record subtyping, is novel. The latter is based on a new encoding of record types. We concentrate on modeling quantifiers and subtyping; recursion is the subject of current work.

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Various formulations of constructive type theories have been proposed to serve as the basis for machine-assisted proof and as a theoretical basis for studying programming languages. Many of these calculi include a cumulative hierarchy of "universes," each a type of types closed under a collection of type-forming operations. Universes are of interest for a variety of reasons, some philosophical (predicative vs. impredicative type theories), some theoretical (limitations on the closure properties of type theories), and some practical (to achieve some of the advantages of a type of all types without sacrificing consistency.) The Generalized Calculus of Constructions (CC ! ) is a formal theory of types that includes such a hierarchy of universes. Although essential to the formalization of constructive mathematics, universes are tedious to use in practice, for one is required to make specific choices of universe levels and to ensure that all choices are consistent. In this pa...

We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as Martin-Lof's type theory [NPS90] and the type theory UTT [Luo94]. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; in particular, subsumption and coercion are combined in such a way that the meaning of an object being in a supertype is given by coercive definition rules for the definitional equality. It is shown that this provides a conceptually simple and uniform framework to understand subtyping and coercion relations in type theories with sophisticated type structures such as inductive types and universes. The use of coercive subtyping in formal development and in reasoning about subsets of objects is discussed in the context of computerassisted formal reasoning. 1 Introduction A type in type theory is often intuitively thought of as a set. For example, types in Martin-Lof's type theory [ML84, NPS90...

In this paper, an extension of Pure Type Systems (PTS's) with definitions is presented. We prove this extension preserves many of the properties of PTS's. The main result is a proof that for many PTS's, including the Calculus of Constructions, this extension preserves strong normalisation.

Constructive type theory as conceived by Per Martin-Löf has a very rich type system, but partial functions cannot be typed. This also makes it impossible to directly write recursive programs. In this paper a constructive type theory Red is defined which includes a partial type constructor A; objects in the type A may diverge, but if they converge, they must be members of A. A fixed point typing principle is given to allow typing of recursive functions. The extraction paradigm of type theory, whereby programs are automatically extracted from constructive proofs, is extended to allow extraction of fixed points. There is a Scott fixed point induction principle for reasoning about these functions. Soundness of the theory is proven. Type theory becomes a more expressive programming logic as a result.

This paper gives a formal description of -- at least a part of -- the type system of Aldor, the extension language of the computer algebra system AXIOM. In the process of doing this a critique of the design of the system emerges.

This paper presents a comparison between algebraic specifications-in-the-large and a type theoretical formulation of modular specifications, called deliverables. It is shown that the laws of module algebra can be translated to laws about deliverables which can be proved correct in type theory. The adequacy of the Extended Calculus of Constructions as a possible implementation of type theory is discussed and it is explained how the reformulation of the laws is influenced by this choice.

We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by Sacerdoti Coen and Zoli in [CSCZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given in [Web91, Web93]. The proof is fully constructive, in the sense that it is done in Bishop’s style and, under certain assumptions, it is also fully predicative. The formalisation is done in the Calculus of (Co)Inductive Constructions using the interactive theorem prover Matita [ASTZ07]. It exploits some peculiar features of Matita and an advanced technique to represent algebraic hierarchies previously introduced by the authors in [ST07]. Moreover, we introduce a new technique to cope with duality to halve the formalisation effort.