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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. 1

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.

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Abstract. Traditionally a rigorous mathematical document consists of a sequence of definition – statement – proof. Taking this basic outline as starting point we investigate how these three categories of text can be represented adequately in the formal language of Isabelle/Isar. Proofs represented in human-readable form have been the initial motivation of Isar language design 10 years ago. The principles developed here allow to turn deductions of the Isabelle logical framework into a format that transcends the raw logical calculus, with more direct description of reasoning using pseudo-natural language elements. Statements describe the main result of a theorem in an open format as a reasoning scheme, saying that in the context of certain parameters and assumptions certain conclusions can be derived. This idea of turning Isar context elements into rule statements has been recently refined to support the dual form of elimination rules as well. Definitions in their primitive form merely name existing elements of the logical environment, by stating a suitable equation or logical equivalence. Inductive definitions provide a convenient derived principle to describe a new predicate as the closure of given natural deduction rules. Again there is a direct connection to Isar principles, rules stemming from an inductive characterization are immediately available in structured reasoning. All three sub-categories benefit from replacing raw logical encodings by native Isar language elements. The overall formality in the presented mathematical text is reduced. Instead of manipulating auxiliary logical connectives and quantifiers, the mathematical concepts are emphasized. 1

A good proof is a proof that makes us wiser. Manin [41, p. 209]. Abstract. Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved. Formal proof is practically achievable. With formal proof, correctness reaches a standard that no pen-and-paper proof can match, but an essential component of mathematics — the insight and understanding — seems to be in short supply. So, what makes a proof understandable? To answer this question we first suggest a list of symptoms of understanding. We then propose a vision of an environment in which users can write and check formal proofs as well as query them with reference to the symptoms of understanding. In this way, the environment reconciles the main features of proof: correctness and understanding. 1

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

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Abstract. 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. 1

Die Dissertation wurde am 02.03.2012 bei der Technischen Universität München This thesis describes work on two components of the interactive theorem prover Isabelle/HOL that generate proofs and counterexamples for higher-order conjectures by harnessing external first-order reasoners. Our primary contribution is the development of Nitpick, a counterexample generator that builds on a first-order relational model finder based on a Boolean satisfiability (SAT) solver. Nitpick supports (co)inductive predicates and datatypes as well as (co)recursive functions. A novel aspect of this work is the use of a monotonicity inference to prune the search space and to soundly interpret infinite types with finite sets, leading to considerable speed and precision improvements. In a case study, Nitpick was successfully applied to an Isabelle formalization of the C++ memory model. Our second main contribution is the further development of the Sledgehammer proof tool. This tool heuristically selects facts relevant to the conjecture to prove,