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A content based mathematical search engine: whelp
 In: Postproceedings of the Types 2004 International Conference, Vol. 3839 of LNCS
, 2004
"... Abstract. The prototype of a content based search engine for mathematical knowledge supporting a small set of queries requiring matching and/or typing operations is described. The prototype — called Whelp — exploits a metadata approach for indexing the information that looks far more flexible than t ..."
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Abstract. The prototype of a content based search engine for mathematical knowledge supporting a small set of queries requiring matching and/or typing operations is described. The prototype — called Whelp — exploits a metadata approach for indexing the information that looks far more flexible than traditional indexing techniques for structured expressions like substitution, discrimination, or context trees. The prototype has been instantiated to the standard library of the Coq proof assistant extended with many user contributions. 1
A LargeScale Experiment in Executing Extracted Programs
"... It is a wellknown fact that algorithms are often hidden inside mathematical proofs. If these proofs are formalized inside a proof assistant, then a mechanism called extraction can generate the corresponding programs automatically. Previous work has focused on the difficulties in obtaining a program ..."
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Cited by 7 (2 self)
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It is a wellknown fact that algorithms are often hidden inside mathematical proofs. If these proofs are formalized inside a proof assistant, then a mechanism called extraction can generate the corresponding programs automatically. Previous work has focused on the difficulties in obtaining a program from a formalization of the Fundamental Theorem of Algebra inside the Coq proof assistant. In theory, this program allows one to compute approximations of roots of polynomials. However, as we show in this work, there is currently a big gap between theory and practice. We study the complexity of the extracted program and analyze the reasons of its inefficiency, showing that this is a direct consequence of the approach used throughout the formalization.
A constructive and formal proof of Lebesgue's Dominated Convergence Theorem in the interactive theorem prover Matita
, 2008
"... We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by Sacerdoti Coen and Zoli in [SZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given in ..."
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Cited by 7 (4 self)
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We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by Sacerdoti Coen and Zoli in [SZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given in [Web91, Web93]. The proof is fully constructive, in the sense that it is done in Bishop’s style and, under certain assumptions, it is also fully predicative. The formalisation is done in the Calculus of (Co)Inductive Constructions using the interactive theorem prover Matita [ASTZ07]. It exploits some peculiar features of Matita and an advanced technique to represent algebraic hierarchies previously introduced by the authors in [ST07]. Moreover, we introduce a new technique to cope with duality to halve the formalisation effort.
The QED Manifesto Revisited
 Studies in Logic, Grammar and Rhetoric
, 2007
"... We present an overview of the current state of formalization of mathematics, and argue what will be needed to make the vision from the QED manifesto come true. ..."
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We present an overview of the current state of formalization of mathematics, and argue what will be needed to make the vision from the QED manifesto come true.
Rough Concept Analysis – theory development in the Mizar system
 Proc. of MKM 2004, Lecture Notes in Computer Science 3119
, 2004
"... Abstract. Theories play an important role in building mathematical knowledge repositories. Organizing knowledge in theories is an obvious approach to cope with the growing number of definitions, theorems, and proofs. However, they are also a matter of subject on their own: developing a new piece of ..."
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Abstract. Theories play an important role in building mathematical knowledge repositories. Organizing knowledge in theories is an obvious approach to cope with the growing number of definitions, theorems, and proofs. However, they are also a matter of subject on their own: developing a new piece of mathematics often relies on extending or combining already developed theories in this way reusing definitions as well as theorems. We believe that this aspect of theory development is crucial for mathematical knowledge management. In this paper we investigate the facilities of the Mizar system concerning extending and combing theories based on structure and attribute definitions. As an example we consider the formation of rough concept analysis out of formal concept analysis and rough sets. 1
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.
Hierarchical Reflection
"... Abstract. The technique of reflection is a way to automate proof construction in type theoretical proof assistants. Reflection is based on the definition of a type of syntactic expressions that gets interpreted in the domain of discourse. By allowing the interpretation function to be partial or even ..."
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Abstract. The technique of reflection is a way to automate proof construction in type theoretical proof assistants. Reflection is based on the definition of a type of syntactic expressions that gets interpreted in the domain of discourse. By allowing the interpretation function to be partial or even a relation one gets a more general method known as ``partial reflection''. In this paper we show how one can take advantage of the partiality of the interpretation to uniformly define a family of tactics for equational reasoning that will work in different algebraic structures. The tactics then follow the hierarchy of those algebraic structures in a natural way.
Engineering Mathematical Knowledge
 Mathematical Knowledge Management, number 3863 in LNAI
, 2005
"... Abstract. Due to their rapidly increasing amount, maintaining mathematical documents more and more becomes an engineering task. In this paper, we combine the projects MMiSS 3 and CDET. 4 That way, we achieve major benefits for mathematical knowledge management: (1) Semantic annotations relate mathem ..."
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Abstract. Due to their rapidly increasing amount, maintaining mathematical documents more and more becomes an engineering task. In this paper, we combine the projects MMiSS 3 and CDET. 4 That way, we achieve major benefits for mathematical knowledge management: (1) Semantic annotations relate mathematical constructs. This reaches beyond mathematics and thus fosters integration of mathematical content into a broader context. (2) Finegrained version control enables change management and configuration management. (3) Semiformal consistency management identifies violations of userdefined consistency requirements and proposes how they can be best resolved. 1
Formalizing Overlap Algebras in Matita
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... We describe some formal topological results, formalized in Matita 1/2, presented in predicative intuitionistic logic and in terms of Overlap Algebras. Overlap Algebras are new algebraic structures designed to ease reasoning about subsets in an algebraic way within intuitionistic logic. We find that ..."
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We describe some formal topological results, formalized in Matita 1/2, presented in predicative intuitionistic logic and in terms of Overlap Algebras. Overlap Algebras are new algebraic structures designed to ease reasoning about subsets in an algebraic way within intuitionistic logic. We find that they also ease the formalization of formal topological results in an interactive theorem prover. Our main result is the existence of a functor between two categories of ‘generalized topological spaces’, one with points (Basic Pairs) and the other pointfree (Basic Topologies). The reported formalization is part as a wider scientific collaboration with the inventor of the theory, Giovanni Sambin. His goal is to verify in what sense, and with what difficulties, his theory is ‘implementable’. We check that all intermediate constructions respect the stringent size requirements imposed by predicative logic. The formalization is quite unusual, since it has to make explicit size information that is often hidden. We found that the version of Matita used for the formalization was largely inappropriate. The formalization drove several major improvements of Matita that will be integrated in the next major release (Matita 1.0). We show some motivating examples for these improvements, taken directly from the formalization. We also describe a possibly suboptimal solution in Matita 1/2, exploitable in other similar systems. We briefly discuss a better solution available in Matita 1.0.
Improving Real Analysis in Coq: a UserFriendly Approach to Integrals and Derivatives ⋆
"... Abstract. Verification of numerical analysis programs requires dealing with derivatives and integrals. High confidence in this process can be achieved using a formal proof checker, such as Coq. Its standard library provides an axiomatization of real numbers and various lemmas about real analysis, wh ..."
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Abstract. Verification of numerical analysis programs requires dealing with derivatives and integrals. High confidence in this process can be achieved using a formal proof checker, such as Coq. Its standard library provides an axiomatization of real numbers and various lemmas about real analysis, which may be used for this purpose. Unfortunately, its definitions of derivative and integral are unpractical as they are partial functions that demand a proof term. This proof term makes the handling of mathematical formulas cumbersome and does not conform to traditional analysis. Other proof assistants usually do not suffer from this issue; for instance, they may rely on Hilbert’s epsilon to get total operators. In this paper, we propose a way to define total operators for derivative and integral without having to extend Coq’s standard axiomatization of real numbers. We proved the compatibility of our definitions with the standard library’s in order to leverage existing results. We also greatly improved automation for real analysis proofs that use Coq standard definitions. We exercised our approach on lemmas involving iterated partial derivatives and differentiation under the integral sign, that were missing from the formal proof of a numerical program solving the wave equation. 1