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. 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 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.

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...

Coercive subtyping is a general approach to subtyping, inheritance and abbreviation in dependent type theories. A vital requirement for coercive subtyping is that of coherence { computational uniqueness of coercions between any two types. In this paper, we develop techniques useful in proving coherence and its related result on admissibility of transitivity and substitution. In particular, we consider suitable subtyping rules for -types and -types and prove its coherence and the admissibility of substitution and transitivity rules at the type level in the coercive subtyping framework. 1

) Zhaohui Luo and Paul Callaghan Department of Computer Science, University of Durham fZhaohui.Luo, P.C.Callaghang@durham.ac.uk 1 Introduction This paper investigates the use of constructive type theory in lexical semantics. Our intention is to explore how a rich language of types with subtyping can be used to express lexical knowledge, both as an application of type theory and as an alternative to current approaches. In particular, we show that coercive subtyping [Luo97, Luo98a], provides a formal framework with useful mechanisms for lexical semantics. Coercive subtyping extends constructive type theories (eg, Martin-Lof's intensional type theory [NPS90] and the type theory UTT [Luo94]) with a simple abbreviational mechanism. It provides elegant and flexible means of representing inheritance and overloading. In our earlier paper on the structure of Mathematical Vernacular [LC98], coercive subtyping is used to represent the inheritance relationships between mathematical concepts and ...

. In the context of Plastic, a proof assistant for a variant of Martin-Lof's Logical Framework LF with explicitly typed -abstractions, we outline the technique used for implementing inductive types from their declarations. This form of inductive types gives rise to a problem of non-linear pattern matching; we propose this match can be ignored in well-typed terms, and outline a proof of this. The paper then explains how the inductive types are realised inside the reduction mechanisms of Plastic, and briefly considers optimisations for inductive types. Key words: type theory, inductive types, LF, implementation. 1 Introduction This paper considers implementation techniques for a particular approach to inductive types in constructive type theory. The inductive types considered are those given in Chapter 9 of [15], in which Luo presents a variant of Martin-Lof's Logical Framework LF which has explicitly typed -abstractions, and a schema for inductive types within this LF which is...

In this paper we present the Durham Mathematical Vernacular (MV) project, discuss the general design of a prototype to support experimentation with issues of MV, explain current work on the prototype -- specifically in the type theory basis of the work, and end with a brief discussion of methodology and future directions. The current work concerns an implementation of Luo's typed logical framework LF, and making it more flexible with respect to the demands of implementing MV -- in particular, meta-variables, multiple contexts, subtyping, and automation. This part of the project may be of particular interest to the general theorem proving community. We will demonstrate a prototype at the conference. 1 Introduction: Defining a Mathematical Vernacular The long term aim of the project is to develop theory and techniques with which the complementary strengths of NLP (Natural Language Processing) and CAFR (Computer-Assisted Formal Reasoning) can be combined to support computer-assisted reas...

Coercive subtyping is a general approach to subtyping, inheritance and abbreviation in dependent type theories. A vital requirement for coercive subtyping is that of coherence which essentially says that coercions between any two types must be unique. Another important task for coercive subtyping is to prove the admissibility or elimination of transitivity and substitution. In this paper, we propose and study the notion of Weak Transitivity, consider suitable subtyping rules for certain parameterised inductive types and prove its coherence and the admissibility of substitution and weak transitivity in the coercive subtyping framework.