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A Compendium of Continuous Lattices in MIZAR  Formalizing recent mathematics
, 2002
"... This paper reports on the Mizar formalization of the theory of continuous lattices as presented in A Compendium of Continuous Lattices, [25]. By the Mizar formalization we mean a formulation of theorems, de nitions, and proofs written in the Mizar language whose correctness is veri ed by the Mizar ..."
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This paper reports on the Mizar formalization of the theory of continuous lattices as presented in A Compendium of Continuous Lattices, [25]. By the Mizar formalization we mean a formulation of theorems, de nitions, and proofs written in the Mizar language whose correctness is veri ed by the Mizar processor. This eort was originally motivated by the question of whether or not the Mizar system was suciently developed for the task of expressing advanced mathematics. The current state of the formalization57 Mizar articles written by 16 authors indicates that in principle the Mizar system has successfully met the challenge. To our knowledge it is the most sizable eort aimed at mechanically checking some substantial and relatively recent eld of advanced mathematics. However, it does not mean that doing mathematics in Mizar is as simple as doing mathematics traditionally (if doing mathematics is simple at all). The work of formalizing the material of [25] has: (i) prompted many improvements of the Mizar proof checking system; (ii) caused numerous revisions of the the Mizar data base; and (iii) contributed to the \to do" list of further changes to the Mizar system.
Proof Assistants: history, ideas and future
"... In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assista ..."
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In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assistants are used and how we envision their extended use in the future. While being an introduction into the world of proof assistants and the main issues behind them, this paper is also a position paper that pushes the further use of proof assistants. We believe that these systems will become the future of mathematics, where definitions, statements, computations and proofs are all available in a computerized form. An important application is and will be in computer supported modelling and verification of systems. But their is still along road ahead and we will indicate what we believe is needed for the further proliferation of proof assistants.
Some considerations on the usability of Interactive Provers
"... Abstract. In spite of the remarkable achievements recently obtained in the field of mechanization of formal reasoning, the overall usability of interactive provers does not seem to be sensibly improved since the advent of the “second generation ” of systems, in the mid of the eighties. We try to ana ..."
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Abstract. In spite of the remarkable achievements recently obtained in the field of mechanization of formal reasoning, the overall usability of interactive provers does not seem to be sensibly improved since the advent of the “second generation ” of systems, in the mid of the eighties. We try to analyze the reasons of such a slow progress, pointing out the main problems and suggesting some possible research directions. 1
Ideas for a MathWiki Editor
"... We present some functional and nonfunctional requirements and wishes for a webbased editor for formalized mathematics, in particular for use in the MathWiki project at RU Nijmegen [13]. We discuss possible implementation alternatives, and argue for a holistic design of the entire wiki with editor ..."
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We present some functional and nonfunctional requirements and wishes for a webbased editor for formalized mathematics, in particular for use in the MathWiki project at RU Nijmegen [13]. We discuss possible implementation alternatives, and argue for a holistic design of the entire wiki with editor features in mind. 1
Theory Presentation Combinators ⋆
"... Abstract. We motivate and give semantics to theory presentation combinators as the foundational building blocks for a scalable library of theories. The key observation is that the category of contexts and fibered categories are the ideal theoretical tools for this purpose. 1 ..."
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Abstract. We motivate and give semantics to theory presentation combinators as the foundational building blocks for a scalable library of theories. The key observation is that the category of contexts and fibered categories are the ideal theoretical tools for this purpose. 1