Sledgehammer, a component of the interactive theorem prover Isabelle, finds proofs in higher-order logic by calling the automated provers for first-order logic E, SPASS and Vampire. This paper is the largest and most detailed empirical evaluation of such a link to date. Our test data consists of 1240 proof goals arising in 7 diverse Isabelle theories, thus representing typical Isabelle proof obligations. We measure the effectiveness of Sledgehammer and many other parameters such as run time and complexity of proofs. A facility for minimizing the number of facts needed to prove a goal is presented and analyzed.

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.

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.

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Formal verification of complex algorithms is challenging. Verifying their implementations goes beyond the state of the art of current automatic verification tools and usually involves intricate mathematical theorems. Certifying algorithms compute in addition to each output a witness certifying that the output is correct.A checker for such a witness is usually much simpler than the original algorithmyet it is all the user has to trust. The verification of checkers is feasible with current tools and leads to computations that can be completely trusted. We describe a framework to seamlessly verify certifying computations. We use the automatic verifier VCC for establishing the correctness of the checker and the interactive theorem prover Isabelle/HOL for high-level mathematical properties of algorithms. We demonstrate the effectiveness of our approach by presenting the verification of typical examples of the industrial-level and widespread algorithmic library LEDA.

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Sledgehammer is a component of the Isabelle/HOL proof assistant that integrates external automatic theorem provers (ATPs) to discharge interactive proof obligations. As a safeguard against bugs, the proofs found by the external provers are reconstructed in Isabelle. Reconstructing complex arguments involves translating them to Isabelle's Isar format, supplying suitable justifications for each step. Sledgehammer transforms the proofs by contradiction into direct proofs; it iteratively tests and compresses the output, resulting in simpler and faster proofs; and it supports a wide range of ATPs, including E, LEO-II, Satallax, SPASS, Vampire, veriT, Waldmeister, and Z3. Keywords Automatic theorem provers • Proof assistants • Natural deduction 1 Introduction Sledgehammer [12,62] is a proof tool that connects the Isabelle/HOL proof assistant [53,54] with external automatic theorem provers (ATPs), including first-order superposition provers, higher-order provers, and solvers based on satisfiability modulo theories (SMT). Given an interactive proof goal, it heuristically selects hundreds of facts (lemmas, definitions, and axioms) from Isabelle's vast libraries, translates them to the external provers' logics, and invokes the provers in parallel (Section 2).