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Mathematical Vernacular and Conceptual Wellformedness in Mathematical Language
 Proceedings of the 2nd Inter. Conf. on Logical Aspects of Computational Linguistics, LNCS/LNAI 1582
, 1998
"... . 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 developmen ..."
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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 wellformedness 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 wellformedness. 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...
ComputerAssisted Mathematics at Work  The HahnBanach Theorem in Isabelle/Isar
 TYPES FOR PROOFS AND PROGRAMS: TYPES’99, LNCS
, 2000
"... We present a complete formalization of the HahnBanach theorem in the simplytyped settheory of Isabelle/HOL, such that both the modeling of the underlying mathematical notions and the full proofs are intelligible to human readers. This is achieved by means of the Isar environment, which provides ..."
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Cited by 7 (4 self)
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We present a complete formalization of the HahnBanach theorem in the simplytyped settheory of Isabelle/HOL, such that both the modeling of the underlying mathematical notions and the full proofs are intelligible to human readers. This is achieved by means of the Isar environment, which provides a framework for highlevel reasoning based on natural deduction. The final result is presented as a readable formal proof document, following usual presentations in mathematical textbooks quite closely. Our case study demonstrates that Isabelle/Isar is capable to support this kind of application of formal logic very well, while being open for an even larger scope.
ComputerAssisted Reasoning with Natural Language: Implementing a Mathematical Vernacular
 In CLUK'98
, 1998
"... This paper presents the Durham Mathematical Vernacular (MV) project. The project aims to develop the technology for interactive tools based on proof checking with type theory, which allow mathematicians to develop mathematics in their usual vernacular. Its subgoals are to develop type theory techno ..."
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This paper presents the Durham Mathematical Vernacular (MV) project. The project aims to develop the technology for interactive tools based on proof checking with type theory, which allow mathematicians to develop mathematics in their usual vernacular. Its subgoals are to develop type theory technology to support MV and to develop the corresponding NL technology. Mathematical language has many significant differences from everyday language, hence a different approach is required to automatically process such language. We discuss some important differences and how they affect implementation. A key requirement is for correctness. Another key feature is that the user defineshis own terminology. We then discuss work in progress, namely the issue of semantic wellformedness in mathematical descriptions, and a prototype being developed to experiment with aspects of the project. 1 Introduction: Defining a Mathematical Vernacular The long term aim of this project is to develop theory and tec...