The DPLL procedure is the basis of some of the most successful propositional satisfiability solvers to date. Although originally devised as a proofprocedure for first-order logic, it has been used almost exclusively for propositional logic so far because of its highly inefficient treatment of quantifiers, based on instantiation into ground formulas. The recent FDPLL calculus by Baumgartner was the first successful attempt to lift the procedure to the first-order level without resorting to ground instantiations. FDPLL lifts to the first-order case the core of the DPLL procedure, the splitting rule, but ignores other aspects of the procedure that, although not necessary for completeness, are crucial for its effectiveness in practice. In this paper, we present a new calculus loosely based on FDPLL that lifts these aspects as well. In addition to being a more faithful litfing of the DPLL procedure, the new calculus contains a more systematic treatment of universal literals, one of FDPLL's optimizations, and so has the potential of leading to much faster implementations.

Irrelevant clauses in resolution problems increase the search space, making it hard to find proofs in a reasonable time. Simple relevance filtering methods, based on counting function symbols in clauses, improve the success rate for a variety of automatic theorem provers and with various initial settings. We have designed these techniques as part of a project to link automatic theorem provers to the interactive theorem prover Isabelle. They should be applicable to other situations where the resolution problems are produced mechanically and where completeness is less important than achieving a high success rate with limited processor time. 1

Abstract. Interactive provers typically use higher-order logic, while automatic provers typically use first-order logic. In order to integrate interactive provers with automatic ones, it is necessary to translate higher-order formulae to first-order form. The translation should ideally be both sound and practical. We have investigated several methods of translating function applications, types and λ-abstractions. Omitting some type information improves the success rate, but can be unsound, so the interactive prover must verify the proofs. This paper presents experimental data that compares the translations in respect of their success rates for three automatic provers. 1.

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Proofs involving large specifications are typically carried out through interactive provers that use higher-order logic. A promising approach to improve the automation of interactive provers is by integrating them with automatic provers, which are usually based on first-order logic. Consequently, it is necessary to translate higher-order logic formulae to first-order form. This translation should ideally be both sound and practical. We have implemented three higher-order to first-order translations, with particular emphasis on the translation of types. Omitting some type information improves the success rate, but can be unsound, so the interactive prover must verify the proofs. In this paper, we will describe our translations and experimental data that compares the three translations in respect of their success rates for various automatic provers. 1

Abstract. We present a new method for automatic generation of loop invariants for programs containing arrays. Unlike all previously known methods, our method allows one to generate first-order invariants containing alternations of quantifiers. The method is based on the automatic analysis of the so-called update predicates of loops. An update predicate for an array A expresses updates made to A. We observe that many properties of update predicates can be extracted automatically from the loop description and loop properties obtained by other methods such as a simple analysis of counters occurring in the loop, recurrence solving and quantifier elimination over loop variables. We run the theorem prover Vampire on some examples and show that non-trivial loop invariants can be generated. 1

Abstract. Interactive proof assistants should verify the proofs they receive from automatic theorem provers. Normally this proof reconstruction takes place internally, forming part of the integration between the two tools. We have implemented source-level proof reconstruction: resolution proofs are automatically translated to Isabelle proof scripts. Users can insert this text into their proof development or (if they wish) examine it manually. Each step of a proof is justified by calling Hurd’s Metis prover, which we have ported to Isabelle. A recurrent issue in this project is the treatment of Isabelle’s axiomatic type classes. 1

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

Abstract. Sledgehammer is a component of Isabelle/HOL that employs firstorder automatic theorem provers (ATPs) to discharge goals arising in interactive proofs. It heuristically selects relevant facts and, if an ATP is successful, produces a snippet that replays the proof in Isabelle. We extended Sledgehammer to invoke satisfiability modulo theories (SMT) solvers as well, exploiting its relevance filter and parallel architecture. Isabelle users are now pleasantly surprised by SMT proofs for problems beyond the ATPs ’ reach. Remarkably, the best SMT solver performs better than the best ATP on most of our benchmarks. 1

We describe feature vector indexing, a new, non-perfect indexing method for clause subsumption. It is suitable for both forward (i.e., finding a subsuming clause in a set) and backward (finding all subsumed clauses in a set) subsumption. Moreover, it is easy to implement, but still yields excellent performance in practice. As an added benefit, by restricting the selection of features used in the index, our technique immediately adapts to indexing modulo arbitrary AC theories with only minor loss of efficiency. Alternatively, the feature selection can be restricted to result in set subsumption. Feature vector indexing has been implemented in our equational theorem prover E, and has enabled us to integrate new simplification techniques making heavy use of subsumption. We experimentally compare the performance of the prover for a number of strategies using feature vector indexing and conventional sequential subsumption. Key words: automated theorem proving, saturation, subsumption, indexing 1