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57
Subtyping Dependent Types
, 2000
"... The need for subtyping in typesystems 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 versio ..."
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Cited by 70 (6 self)
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The need for subtyping in typesystems 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 typechecking.
User Interaction with the Matita Proof Assistant
 Journal of Automated Reasoning, Special
, 2006
"... Abstract. Matita is a new, documentcentric, tacticbased 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 highquality not ..."
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Cited by 51 (15 self)
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Abstract. Matita is a new, documentcentric, tacticbased 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 highquality notational rendering, and the complex interplay between syntax, presentation, and semantics.
Implicit Coercions in Type Systems
 In Selected Papers from the International Workshop TYPES '95
, 1995
"... . 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 ..."
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Cited by 17 (1 self)
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. 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 \Pitypes. 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...
An Implementation of LF with Coercive Subtyping & Universes
 Journal of Automated Reasoning
"... . We present `Plastic', an implementation of LF with Coercive Subtyping, and focus on its implementation of Universes. LF is a variant of MartinLof's logical framework, with explicitly typed abstractions. We outline the system of LF with its extensions of inductive types and coercions. ..."
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Cited by 15 (9 self)
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. We present `Plastic', an implementation of LF with Coercive Subtyping, and focus on its implementation of Universes. LF is a variant of MartinLof'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 Tarskistyle universes together with coercive subtyping provides an ideal formulation of universes which is both semantically clear and practical to use. Keywords: type theory, un...
Coercion Completion and Conservativity in Coercive Subtyping
 ANNALS OF PURE AND APPLIED LOGIC
, 2000
"... 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 suc ..."
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Cited by 15 (9 self)
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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 MartinLof's type theory and UTT is a conservative extension. The importance of coherence conditions for the conservativity result is also discussed.
Dependently Typed Records for Representing Mathematical Structure
 Theorem Proving in Higher Order Logics, TPHOLs 2000
, 2000
"... this paper appears in Theorem Proving in Higher Order Logics, TPHOLs 2000, c ..."
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this paper appears in Theorem Proving in Higher Order Logics, TPHOLs 2000, c
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...
The algebraic hierarchy of the FTA Project
 Journal of Symbolic Computation, Special Issue on the Integration of Automated Reasoning and Computer Algebra Systems
, 2002
"... 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 her ..."
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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 nontrivial mathematics as part of the FTA project.
A Theory of Typed Coercions and its Applications
"... A number of important program rewriting scenarios can be recast as typedirected coercion insertion. These range from more theoretical applications such as coercive subtyping and supporting overloading in type theories, to more practical applications such as integrating static and dynamically typed ..."
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Cited by 13 (4 self)
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A number of important program rewriting scenarios can be recast as typedirected coercion insertion. These range from more theoretical applications such as coercive subtyping and supporting overloading in type theories, to more practical applications such as integrating static and dynamically typed code using gradual typing, and inlining code to enforce security policies such as access control and provenance tracking. In this paper we give a general theory of typedirected coercion insertion. We specifically explore the inherent tradeoff between expressiveness and ambiguity—the more powerful the strategy for generating coercions, the greater the possibility of several, semantically distinct rewritings for a given program. We consider increasingly powerful coercion generation strategies, work out example applications supported by the increased power (including those mentioned above), and identify the inherent ambiguity problems of each setting, along with various techniques to tame the ambiguities.