In the recent years, the Metis prover based on ordered paramodulation and model elimination has replaced the earlier built-in methods for general-purpose proof automation in HOL4 and Isabelle HOL. In the annual CASC competition, the leanCoP system based on connection tableaux has however performed better than Metis. In this paper we show how the leanCoP's core algorithm can be implemented inside HOLLight. leanCoP's flagship feature, namely its minimalistic core, results in a very simple proof system. This plays a crucial role in extending the MESON proof reconstruction mechanism to connection tableaux proofs, providing an implementation of leanCoP that certifies its proofs. We discuss the differences between our direct implementation using an explicit Prolog stack, to the continuation passing implementation of MESON present in HOLLight and compare their performance on all core HOLLight goals. The resulting prover can be also used as a general purpose TPTP prover. We compare its performance against the resolution based Metis on TPTP and other interesting datasets.
  In this paper we give a short introduction in first-order theorem proving and the use of the theorem prover Vampire. We discuss the superposition calculus and explain the key concepts of saturation and redundancy elimination, present saturation algorithms and preprocessing, and demonstrate how these concepts are implemented in Vampire. Further, we also cover more recent topics and features of Vampire designed for advanced applications, including satisfiability checking, theory reasoning, interpolation, consequence elimination, and program analysis.  In this paper, an overview of state-of-the-art techniques for premise selection in large theory mathematics is provided, and new premise selection techniques are introduced. Several evaluation metrics are introduced, compared and their appropriateness is discussed in the context of automated reasoning in large theory mathematics. The methods are evaluated on the MPTP2078 benchmark, a subset of the Mizar library, and a 10 improvement is obtained over the best method so far. 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 "zero-click" 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. 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. A proof procedure based on a theorem of Herbrand and utilizing the matching technique of Prawitz is presented. In general, Herbrand-type proof procedures proceed by generating over increasing numbers of candidates for the truth-functionally contradictory statement the procedures seek. A trial is successful when some candidate is in fact a contradictory statement. In procedures to date the number of candidates developed before a contradictory statement is found (if one is found) varies roughly exponentially with the size of the contradictory statement. (“Size” might be measured by the number of clauses in the conjunctive normal form of the contradictory statement.) Although basically subject to the same rate of growth, the procedure introduced here attempts to drastically trim the number of candidates at an intermediate level of development. This is done by retaining beyond a certain level only candidates already “partially contradictory.” The major task usually is finding the partially contradictory sets. However, the number of candidate sets required to find these subsets of the contradictory set is generally much smaller than the number required to find the full contradictory set. HOL(y)Hammer is an online AI ATP service for formal (computer-understandable) mathematics encoded in the HOL Light system. The service allows its users to upload and automatically process an arbitrary formal development (project) based on HOL Light, and to attack arbitrary conjectures that use the concepts defined in some of the uploaded projects. For that, the service uses several automated reasoning systems combined with several premise selection methods trained on all the project proofs. The projects that are readily available on the server for such query answering include the recent versions of the Flyspeck, Multivariate Analysis and Complex Analysis libraries. The service runs on a 48-CPU server, currently employing in parallel for each task 7 AI ATP combinations and 4 decision procedures that contribute to its overall performance. The system is also available for local installation by interested users, who can customize it for their own proof development. An Emacs interface allowing parallel asynchronous queries to the service is also provided. The overall structure of the service is outlined, problems that arise and their solutions are discussed, and an initial account of using the system is given. E is an equational theorem prover for clausal logic with equality. We describe the latest version, E 0.81 Tumsong, with special emphasis on the important aspects that have changed compared to previously described versions. A generic tableau prover has been implemented and integrated with Isabelle (Paulson, 1994). Compared with classical first-order logic provers, it has numerous extensions that allow it to reason with any supplied set of tableau rules. It has a higherorder syntax in order to support user-defined binding operators, such as those of set theory. The unification algorithm is first-order instead of higher-order, but it includes modifications to handle bound variables. The proof, when found, is returned to Isabelle as a list of tactics. Because Isabelle verifies the proof, the prover can cut corners for efficiency’s sake without compromising soundness. For example, the prover can use type information to guide the search without storing type information in full. Categories: F.4, I.1 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. PRocH is a proof reconstruction tool that imports in HOL Light proofs produced by ATPs on the recently developed translation of HOL Light and Flyspeck problems to ATP formats. PRocH combines several reconstruction methods in parallel, but the core improvement over previous methods is obtained by re-playing in the HOL logic the detailed inference steps recorded in the ATP (TPTP) proofs, using several internal HOL Light inference methods. These methods range from fast variable matching and more involved rewriting, to full first-order theorem proving using the MESON tactic. The system is described and its performance is evaluated here on a large set of Flyspeck problems.  Many implementations of model elimination perform proof search by iteratively increasing a bound on the total size of the proof. We propose an optimized version of this search mode using a simple divide-and-conquer refinement. Optimized and unoptimized modes are compared, together with depth-bounded and best-first search, over the entire TPTP problem library. The optimized size-bounded mode seems to be the overall winner, but for each strategy there are problems on which it performs best. Some attempt is made to analyze why. We emphasize that our optimization, and other implementation techniques like caching, are rather general: they are not dependent on the details of model elimination, or even that the search is concerned with theorem proving. As such, we believe that this study is a useful complement to research on extending the model elimination calculus.