The need for subtyping in type-systems with dependent types has been realized for some years. But it is hard to prove that systems combining the two features have fundamental properties such as subject reduction. Here we investigate a subtyping extension of the system *P, which is an abstract version of the type system of the Edinburgh Logical Framework LF. By using an equivalent formulation, we establish some important properties of the new system *P^, including subject reduction. Our analysis culminates in a complete and terminating algorithm which establishes the decidability of type-checking.

Abstract. Matita is a new, document-centric, tactic-based interactive theorem prover. This paper focuses on some of the distinctive features of the user interaction with Matita, mostly characterized by the organization of the library as a searchable knowledge base, the emphasis on a high-quality notational rendering, and the complex interplay between syntax, presentation, and semantics.

Abstract. We describe a framework for algebraic expressions for the proof assistant Coq. This framework has been developed as part of the FTA project in Nijmegen, in which a complete proof of the fundamental theorem of algebra has been formalized in Coq. The algebraic framework that is described here is both abstract and structured. We apply a combination of record types, coercive subtyping and implicit arguments. The algebraic framework contains a full development of the real and complex numbers and of the rings of polynomials over these fields. The framework is constructive. It does not use anything apart from the Coq logic. The framework has been successfully used to formalize non-trivial mathematics as part of the FTA project.

this paper appears in Theorem Proving in Higher Order Logics, TPHOLs 2000, c

. This paper investigates the semantics of mathematical concepts in a type theoretic framework with coercive subtyping. The typetheoretic analysis provides a formal semantic basis in the design and implementation of Mathematical Vernacular (MV), a natural language suitable for interactive development of mathematics with the support of the current theorem proving technology. The idea of semantic well-formedness in mathematical language is motivated with examples. A formal system based on a notion of conceptual category is then presented, showing how type checking supports our notion of well-formedness. The power of this system is then extended by incorporating a notion of subcategory, using ideas from a more general theory of coercive subtyping, which provides the mechanisms for modelling conventional abbreviations in mathematics. Finally, we outline how this formal work can be used in an implementation of MV. 1 Introduction By mathematical vernacular (MV), we mean a mathematical and n...

. We propose a notion of pure type system with implicit coercions. In our framework, judgements are extended with a context of coercions \Delta and the application rule is modified so as to allow coercions to be left implicit. The setting supports multiple inheritance and can be applied to all type theories with \Pi-types. One originality of our work is to propose a computational interpretation for implicit coercions. In this paper, we demonstrate how this interpretation allows a strict control on the logical properties of pure type systems with implicit coecions. 1 Introduction The increasing importance of mathematical software has been accompanied by a drift of mainstream mathematics towards mathematical logic and the foundations of mathematics. Before mathematical software, formal systems were generally seen both by logicians and mathematicians as safe heavens into which mathematics could theoretically be embedded. With powerful mathematical software, there is now a genuine interes...

. We present `Plastic', an implementation of LF with Coercive Subtyping, and focus on its implementation of Universes. LF is a variant of Martin-Lof's logical framework, with explicitly typed -abstractions. We outline the system of LF with its extensions of inductive types and coercions. Plastic is the first implementation of this extended system; we discuss motivations and basic architecture, and give examples of its use. LF is used to specify type theories. The theory UTT includes a hierarchy of universes which is specified in Tarski style. We outline the theory of these universes and explain how they are implemented in Plastic. Of particular interest is the relationship between universes and inductive types, and the relationship between universes and coercive subtyping. We claim that the combination of Tarski-style universes together with coercive subtyping provides an ideal formulation of universes which is both semantically clear and practical to use. Keywords: type theory, un...

Coercive subtyping offers a general approach to subtyping and inheritance by introducing a simple abbreviational mechanism to constructive type theories. In this paper, we study coercion completion in coercive subtyping and prove that the formal extension with coercive subtyping of a type theory such as Martin-Lof's type theory and UTT is a conservative extension. The importance of coherence conditions for the conservativity result is also discussed.

We describe a framework of algebraic structures in the proof assistant Coq. We have developed this framework as part of the FTA project in Nijmegen, in which a constructive proof of the Fundamental Theorem of Algebra has been formalized in Coq. The algebraic hierarchy that is described here is both abstract and way, dening e.g. a ring as a tuple consisting of a group, a binary operation and a constant that together satisfy the properties of a ring. In this way, a ring automatically inherits the group properties of the additive subgroup. The algebraic hierarchy is formalized in Coq by applying a combination of labeled record types and coercions. In the labeled record types of Coq, one can use dependent types: the type of one label may depend on another label. This allows to give a type to a dependent-typed tuple like hA; f; ai, where A is a set, f an operation on A and a an element of A. Coercions are

A notion of dependent coercion is introduced and studied in the context of dependent type theories. It extends our earlier work on coercive subtyping into a uniform framework which increases the expressive power with new applications. A dependent coercion introduces a subtyping relation between a type and a family of types in that an object of the type is mapped into one of the types in the family. We present the formal framework, discuss its meta-theory, and consider applications such as its use in functional programming with dependent types. 1 Introduction Coercive subtyping, as studied in [Luo97, Luo99, JLS98], represents a new general approach to subtyping and inheritance in type theory. In particular, it provides a framework in which subtyping, inheritance, and abbreviation can be understood in dependent type theories where types are understood as consisting of canonical objects. In this paper, we extend the framework of coercive subtyping to introduce a notion of dependent coer...