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Hammering towards QED
"... This paper surveys the emerging methods to automate reasoning over large libraries developed with formal proof assistants. We call these methods hammers. They give the authors of formal proofs a strong “onestroke ” tool for discharging difficult lemmas without the need for careful and detailed manu ..."
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This paper surveys the emerging methods to automate reasoning over large libraries developed with formal proof assistants. We call these methods hammers. They give the authors of formal proofs a strong “onestroke ” tool for discharging difficult lemmas without the need for careful and detailed manual programming of proof search. The main ingredients underlying this approach are efficient automatic theorem provers that can cope with hundreds of axioms, suitable translations of the proof assistant’s logic to the logic of the automatic provers, heuristic and learning methods that select relevant facts from large libraries, and methods that reconstruct the automatically found proofs inside the proof assistants. We outline the history of these methods, explain the main issues and techniques, and show their strength on several large benchmarks. We also discuss the relation of this technology to the QED Manifesto and consider its implications for QEDlike efforts. 1.
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.
Xmonad in Coq (experience report): Programming a window manager in a proof assistant
 In Haskell Symposium
, 2012
"... This report documents the insights gained from implementing the core functionality of xmonad, a popular window manager written in Haskell, in the Coq proof assistant. Rather than focus on verification, this report outlines the technical challenges involved with incorporating Coq code in a Haskell p ..."
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This report documents the insights gained from implementing the core functionality of xmonad, a popular window manager written in Haskell, in the Coq proof assistant. Rather than focus on verification, this report outlines the technical challenges involved with incorporating Coq code in a Haskell project.
Equational Reasoning in Algebraic Structures: a Complete Tactic
"... We present rational, a Coq tactic for equational reasoning in abelian groups, commutative rings, and fields. We give an mathematical description of the method that this tactic uses, which abstracts from Coq specifics. We prove that the method that rational uses is correct, and that it is complete fo ..."
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We present rational, a Coq tactic for equational reasoning in abelian groups, commutative rings, and fields. We give an mathematical description of the method that this tactic uses, which abstracts from Coq specifics. We prove that the method that rational uses is correct, and that it is complete for groups and rings. Completeness means that the method succeeds in proving an equality if and only if that equality is provable from the the group/ring axioms. Finally we characterize in what way our method is incomplete for fields.
Hybrid System verification in Coq
"... Abstract. This internship is intended to improve the abstraction method described by Alur in [2], and implemented in Coq in Nimegen [5,7] for proving the safety of hybrid systems. 1 ..."
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Abstract. This internship is intended to improve the abstraction method described by Alur in [2], and implemented in Coq in Nimegen [5,7] for proving the safety of hybrid systems. 1
Position paper: A real Semantic Web for mathematics deserves a real semantics
"... Abstract. Mathematical documents, and their instrumentation by computers, have rich structure at the layers of presentation, metadata and semantics, as objects in a system for formal mathematical logic. Semantic Web tools [2] support the first two of these, with little, if any, contribution to the t ..."
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Abstract. Mathematical documents, and their instrumentation by computers, have rich structure at the layers of presentation, metadata and semantics, as objects in a system for formal mathematical logic. Semantic Web tools [2] support the first two of these, with little, if any, contribution to the third, while Proof Assistants [17] instrument the third layer, typically with bespoke approaches to the first two. Our position is that a web of mathematical documents, definitions and proofs should be given a fullyfledged semantics in terms of the third layer. We propose a “MathWiki ” to harness Web 2.0 tools and techniques to the rich semantics furnished by contemporary Proof Assistants. 1 Background and state of the art We can identify four worlds of mathematical discourse available on the Web: – Traditional mathematical practice: a systematic body of knowledge, organised around documents written by experts, most often in L ATEX, to varying degrees of sophistication. The intended audience is an expert readership, and
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"... We present rational, a Coq tactic for equational reasoning in abelian groups, commutative rings, and fields. We give an mathematical description of the method that this tactic uses, which abstracts from Coq specifics. We prove that the method that rational uses is correct, and that it is complete fo ..."
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We present rational, a Coq tactic for equational reasoning in abelian groups, commutative rings, and fields. We give an mathematical description of the method that this tactic uses, which abstracts from Coq specifics. We prove that the method that rational uses is correct, and that it is complete for groups and rings. Completeness means that the method succeeds in proving an equality if and only if that equality is provable from the the group/ring axioms. Finally we characterize in what way our method is incomplete for fields.
J Autom Reasoning (2007) 39:109–139 DOI 10.1007/s1081700790705 User Interaction with the Matita Proof Assistant
"... 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, characterized mostly by the organization of the library as a searchable knowledge base, the emphasis on a highquality not ..."
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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, characterized mostly 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.