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A module calculus for Pure Type Systems
, 1996
"... Several proofassistants rely on the very formal basis of Pure Type Systems. However, some practical issues raised by the development of large proofs lead to add other features to actual implementations for handling namespace management, for developing reusable proof libraries and for separate verif ..."
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Cited by 23 (3 self)
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Several proofassistants rely on the very formal basis of Pure Type Systems. However, some practical issues raised by the development of large proofs lead to add other features to actual implementations for handling namespace management, for developing reusable proof libraries and for separate verification of distincts parts of large proofs. Unfortunately, few theoretical basis are given for these features. In this paper we propose an extension of Pure Type Systems with a module calculus adapted from SMLlike module systems for programming languages. Our module calculus gives a theoretical framework addressing the need for these features. We show that our module extension is conservative, and that type inference in the module extension of a given PTS is decidable under some hypotheses over the considered PTS.
Axiomatic constructor classes in Isabelle/HOLCF
 In In Proc. 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs ’05), Volume 3603 of Lecture Notes in Computer Science
, 2005
"... Abstract. We have definitionally extended Isabelle/HOLCF to support axiomatic Haskellstyle constructor classes. We have subsequently defined the functor and monad classes, together with their laws, and implemented state and resumption monad transformers as generic constructor class instances. This ..."
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Cited by 16 (5 self)
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Abstract. We have definitionally extended Isabelle/HOLCF to support axiomatic Haskellstyle constructor classes. We have subsequently defined the functor and monad classes, together with their laws, and implemented state and resumption monad transformers as generic constructor class instances. This is a step towards our goal of giving modular denotational semantics for concurrent lazy functional programming languages, such as GHC Haskell. 1
STMM: A Set Theory for Mechanized Mathematics
 JOURNAL OF AUTOMATED REASONING
, 2000
"... Although set theory is the most popular foundation for mathematics, not many mechanized mathematics systems are based on set theory. ZermeloFraenkel (zf) set theory and other traditional set theories are not an adequate foundation for mechanized mathematics. stmm is a version of vonNeumannBerna ..."
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Cited by 12 (6 self)
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Although set theory is the most popular foundation for mathematics, not many mechanized mathematics systems are based on set theory. ZermeloFraenkel (zf) set theory and other traditional set theories are not an adequate foundation for mechanized mathematics. stmm is a version of vonNeumannBernaysGödel (nbg) set theory that is intended to be a Set Theory for Mechanized Mathematics. stmm allows terms to denote proper classes and to be undened, has a denite description operator, provides a sort system for classifying terms by value, and includes lambdanotation with term constructors for function application and function abstraction. This paper describes stmm and discusses why it is a good foundation for mechanized mathematics.
An Overview of A Formal Framework For Managing Mathematics
 Annals of Mathematics and Artificial Intelligence
, 2003
"... Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform t ..."
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Cited by 12 (6 self)
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Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform theory which is simultaneously an axiomatic theory and an algorithmic theory. Representing a collection of mathematical models, a biform theory provides a formal context for both deduction and computation. The framework includes facilities for deriving theorems via a mixture of deduction and computation, constructing sound deduction and computation rules, and developing networks of biform theories linked by interpretations. The framework is not tied to a specific underlying logic; indeed, it is intended to be used with several background logics simultaneously. Many of the ideas and mechanisms used in the framework are inspired by the imps Interactive Mathematical Proof System and the Axiom computer algebra system.
A Set Theory with Support for Partial Functions
 STUDIA LOGICA
, 2000
"... Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its dom ..."
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Cited by 9 (2 self)
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Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambdanotation and sorts or types as in type theory. This paper introduces a version of vonNeumannBernaysGödel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial firstorder logic, so classvalued terms may be nondenoting. Functions can be specified using lambdanotation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the imps Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.
A Basic Extended Simple Type Theory
, 2001
"... This paper presents an extended version of Church's simple type theory called Basic Extended Simple Type Theory (bestt). By adding type variables and support for reasoning with tuples, lists, and sets to simple type theory, it is intended to be a practical logic for formalized mathematics. 1 ..."
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Cited by 2 (1 self)
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This paper presents an extended version of Church's simple type theory called Basic Extended Simple Type Theory (bestt). By adding type variables and support for reasoning with tuples, lists, and sets to simple type theory, it is intended to be a practical logic for formalized mathematics. 1
An Exploration Tool for Formal Proofs
"... Proof assistants aid the user in proving mathematical theorems by taking care of lowlevel reasoning details. Their user interfaces often present proof information as text, which becomes increasingly difficult to comprehend as it grows in size. Panoptes is a software tool that enables users to explo ..."
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Proof assistants aid the user in proving mathematical theorems by taking care of lowlevel reasoning details. Their user interfaces often present proof information as text, which becomes increasingly difficult to comprehend as it grows in size. Panoptes is a software tool that enables users to explore graphical representations of the formal proofs produced by the imps Interactive Mathematical Proof System. Panoptes automatically displays an imps deduction graph as a visual graph that can be easily manipulated by the user. Its facilities include target zooming, floating information boxes, node relabeling, and proper substructure collapsing.
iv Table of Contents
"... Many issues stand in the way of the development of contemporary mechanized mathematics systems (MMS) and the following are two major obstacles: • Dedicated languages with mathematical specifications for MMS. • A wellendowed theory library which serves as a database of mathematics. We implement a Ma ..."
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Many issues stand in the way of the development of contemporary mechanized mathematics systems (MMS) and the following are two major obstacles: • Dedicated languages with mathematical specifications for MMS. • A wellendowed theory library which serves as a database of mathematics. We implement a MathScheme Language (MSL), which represents theory types useful for covering basic algebraic structures and improving the expressive power of mathematical modeling systems. The development of MSL primarily focuses on language syntax and its logic independence. More importantly, we present a library of theory types developed based on module systems of typed programming languages and algebraic specification languages. The modularity mechanism used in our library aims for the interface manipulation and high level expressivity of MMSs. The theories are organized according to the little theories method [10]. Our module system extensively supports several building operations to construct new theories from existing theories. iii Acknowledgements I would first and foremost like to express my deepfelt gratitude to my supervisor, Dr. Jacques Carette, who shared with me a lot of his expertise and research insight through my studies. This thesis was made possible by his advice, assistance and guidance. My special thanks and appreciation goes to the members of my examination committee, Dr. Jacques Carette, Dr. William M. Farmer, and Dr. Spencer