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Three Years of Experience with Sledgehammer, a Practical Link between Automatic and Interactive Theorem Provers
"... Sledgehammer is a highly successful subsystem of Isabelle/HOL that calls automatic theorem provers to assist with interactive proof construction. It requires no user configuration: it can be invoked with a single mouse gesture at any point in a proof. It automatically finds relevant lemmas from all ..."
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Sledgehammer is a highly successful subsystem of Isabelle/HOL that calls automatic theorem provers to assist with interactive proof construction. It requires no user configuration: it can be invoked with a single mouse gesture at any point in a proof. It automatically finds relevant lemmas from all those currently available. An unusual aspect of its architecture is its use of unsound translations, coupled with its delivery of results as Isabelle/HOL proof scripts: its output cannot be trusted, but it does not need to be trusted. Sledgehammer works well with Isar structured proofs and allows beginners to prove challenging theorems.
M.: Authoring verified documents by interactive proof construction and verification in texteditors
 Lecture Notes in Computer Science Volume 5144
, 2008
"... Abstract. Aiming at a documentcentric approach to formalizing and verifying mathematics and software we integrated the proof assistance system ΩMEGA with the standard scientific texteditor TEXMACS. The author writes her mathematical document entirely inside the texteditor in a controlled language ..."
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Abstract. Aiming at a documentcentric approach to formalizing and verifying mathematics and software we integrated the proof assistance system ΩMEGA with the standard scientific texteditor TEXMACS. The author writes her mathematical document entirely inside the texteditor in a controlled language with formulas in LATEX style. The notation specified in such a document is used for both parsing and rendering formulas in the document. To make this approach effectively usable as a realtime application we present an efficient hybrid parsing technique that is able to deal with the scalability problem resulting from modifying or extending notation dynamically. Furthermore, we present incremental methods to quickly verify constructed or modified proof steps by ΩMEGA. If the system detects incomplete or underspecified proof steps, it tries to automatically repair them. For collaborative authoring we propose to manage partially or fully verified documents together with its justifications and notational information centrally in a mathematics repository using an extension of OMDOC. 1
Automatic Proof and Disproof in Isabelle/HOL
, 2011
"... Isabelle/HOL is a popular interactive theorem prover based on higherorder logic. It owes its success to its ease of use and powerful automation. Much of the automation is performed by external tools: The metaprover Sledgehammer relies on resolution provers and SMT solvers for its proof search, the c ..."
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Cited by 3 (1 self)
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Isabelle/HOL is a popular interactive theorem prover based on higherorder logic. It owes its success to its ease of use and powerful automation. Much of the automation is performed by external tools: The metaprover Sledgehammer relies on resolution provers and SMT solvers for its proof search, the counterexample generator Quickcheck uses the ML compiler as a fast evaluator for ground formulas, and its rival Nitpick is based on the model finder Kodkod, which performs a reduction to SAT. Together with the Isar structured proof format and a new asynchronous user interface, these tools have radically transformed the Isabelle user experience. This paper provides an overview of the main automatic proof and disproof tools.
Sharedmemory multiprocessing for interactive theorem proving
 Interactive Theorem Proving  4th International Conference, ITP 2013
"... Abstract. We address the multicore problem for interactive theorem proving, notably for Isabelle. The stagnation of CPU clock frequency since 2005 means that hardware manufactures multiply cores to keep up with “Moore’s Law”, but this imposes the burden of explicit parallelism to application develop ..."
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Abstract. We address the multicore problem for interactive theorem proving, notably for Isabelle. The stagnation of CPU clock frequency since 2005 means that hardware manufactures multiply cores to keep up with “Moore’s Law”, but this imposes the burden of explicit parallelism to application developers. To cope with this trend, Isabelle has started to support parallel theory and proof processing in 2007, and continuously improved the use of multicore hardware in recent years. This is of practical relevance to theory and proof development, since their size and complexity is roughly correlated with the real time required for rechecking. Scaling up the prover on parallel hardware will facilitate maintenance of larger theory libraries, for example. Our approach to parallel processing in Isabelle is mostly implicit, without user intervention. The system is able to exploit the inherent problemstructure of LCFstyle proof checking, although it requires substantial reforms of the prover architecture and its implementation. Thus the user gains significant speedup factors on typical commodity hardware with 2–32 cores; saturation of 8 cores is already routine in many applications. The present paper provides an overview of the current state of sharedmemory multiprocessing in Isabelle2013, which also benefits from recent improvements of parallel memory management in Poly/ML (by David Matthews). We discuss common requirements, problems, and solutions. Concrete performance figures are analyzed for some applications from the Isabelle distribution and the Archive of Formal Proofs (AFP).
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.
Redirecting proofs by contradiction
"... This paper presents an algorithm that redirects proofs by contradiction. The input is a refutation graph, as produced by an automatic theorem prover (e.g., E, SPASS, Vampire, Z3); the output is a direct proof expressed in natural deduction extended with case analyses and nested subproofs. The algori ..."
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This paper presents an algorithm that redirects proofs by contradiction. The input is a refutation graph, as produced by an automatic theorem prover (e.g., E, SPASS, Vampire, Z3); the output is a direct proof expressed in natural deduction extended with case analyses and nested subproofs. The algorithm is implemented in Isabelle’s Sledgehammer, where it enhances the legibility of machinegenerated proofs.
Robust, SemiIntelligible Isabelle Proofs from ATP Proofs
"... Sledgehammer integrates external automatic theorem provers (ATPs) in the Isabelle/HOL proof assistant. To guard against bugs, ATP proofs must be reconstructed in Isabelle. Reconstructing complex proofs involves translating them to detailed Isabelle proof texts, using suitable proof methods to justif ..."
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Sledgehammer integrates external automatic theorem provers (ATPs) in the Isabelle/HOL proof assistant. To guard against bugs, ATP proofs must be reconstructed in Isabelle. Reconstructing complex proofs involves translating them to detailed Isabelle proof texts, using suitable proof methods to justify the inferences. This has been attempted before with little success, but we have addressed the main issues: Sledgehammer now transforms the proofs by contradiction into direct proofs (as described in a companion paper [3]); it reconstructs skolemization inferences correctly; it provides the right amount of type annotations to ensure formulas are parsed correctly without marring them with types; and it iteratively tests and compresses the output, resulting in simpler and faster working proofs.
MaSh: Machine Learning for Sledgehammer
"... Abstract. Sledgehammer integrates automatic theorem provers in the proof assistant Isabelle/HOL. A key component, the relevance filter, heuristically ranks the thousands of facts available and selects a subset, based on syntactic similarity to the current goal. We introduce MaSh, an alternative that ..."
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Abstract. Sledgehammer integrates automatic theorem provers in the proof assistant Isabelle/HOL. A key component, the relevance filter, heuristically ranks the thousands of facts available and selects a subset, based on syntactic similarity to the current goal. We introduce MaSh, an alternative that learns from successful proofs. New challenges arose from our “zeroclick ” vision: MaSh should integrate seamlessly with the users ’ workflow, so that they benefit from machine learning without having to install software, set up servers, or guide the learning. The underlying machinery draws on recent research in the context of Mizar and HOL Light, with a number of enhancements. MaSh outperforms the old relevance filter on large formalizations, and a particularly strong filter is obtained by combining the two filters. 1
Premise Selection in the Naproche System
"... Abstract. Automated theorem provers (ATPs) struggle to solve problems with large sets of possibly superfluous axiom. Several algorithms have been developed to reduce the number of axioms, optimally only selecting the necessary axioms. However, most of these algorithms consider only single problems. ..."
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Abstract. Automated theorem provers (ATPs) struggle to solve problems with large sets of possibly superfluous axiom. Several algorithms have been developed to reduce the number of axioms, optimally only selecting the necessary axioms. However, most of these algorithms consider only single problems. In this paper, we describe an axiom selection method for series of related problems that is based on logical and textual proximity and tries to mimic a human way of understanding mathematical texts. We present first results that indicate that this approach is indeed useful. Key words: formal mathematics, automated theorem proving, axiom selection 1