This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project.

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the Flyspeck proofs, producing an AI system capable of proving a wide range of mathematical conjectures automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39 % of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided.

As a present to Mizar on its 40th anniversary, we develop an AI/ATP system that in 30 seconds of real time on a 14-CPU machine automatically proves 40% of the theorems in the latest official version of the Mizar Mathematical Library (MML). This is a considerable improvement over previous performance of large- theory AI/ATP methods measured on the whole MML. To achieve that, a large suite of AI/ATP methods is employed and further developed. We implement the most useful methods efficiently, to scale them to the 150000 formulas in MML. This reduces the training times over the corpus to 1-3 seconds, allowing a simple practical deployment of the methods in the online automated reasoning service for the Mizar users (MizAR)

Deep learning techniques lie at the heart of several significant AI advances in recent years including object recognition and detection, image captioning, machine translation, speech recognition and synthesis, and playing the game of Go.Automated first-order theorem provers can aid in the formalization and verification of mathematical theorems and play a crucial role in program analysis, theory reasoning, security, interpolation, and system verification.Here we suggest deep learning based guidance in the proof search of the theorem prover E. We train and compare several deep neural network models on the traces of existing ATP proofs of Mizar statements and use them to select processed clauses during proof search. We give experimental evidence that with a hybrid, two-phase approach, deep learning based guidance can significantly reduce the average number of proof search steps while increasing the number of theorems proved.Using a few proof guidance strategies that leverage deep neural networks, we have found first-order proofs of 7.36% of the first-order logic translations of the Mizar Mathematical Library theorems that did not previously have ATP generated proofs. This increases the ratio of statements in the corpus with ATP generated proofs from 56% to 59%.

Sledgehammer integrates automatic theorem provers in the proof assistant Isabelle/HOL. A key component, the fact selector, heuristically ranks the thousands of facts (lemmas, definitions, or axioms) available and selects a subset, based on syntactic similarity to the current proof goal. We introduce MaSh, an alternative that learns from successful proofs. New challenges arose from our "zero click" vision: MaSh integrates seamlessly with the users' workflow, so that they benefit from machine learning without having to install software, set up servers, or guide the learning. MaSh outperforms the old fact selector on large formalizations.

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.

Hammers provide most powerful general purpose automation for proof assistants based on HOL and set theory today. Despite the gaining popularity of the more advanced versions of type theory, such as those based on the Calculus of Inductive Constructions, the construction of hammers for such foundations has been hindered so far by the lack of translation and reconstruction components. In this paper, we present an architecture of a full hammer for dependent type theory together with its implementation for the Coq proof assistant. A key component of the hammer is a proposed translation from the Calculus of Inductive Constructions, with certain extensions introduced by Coq, to untyped first-order logic. The translation is “sufficiently” sound and complete to be of practical use for automated theorem provers. We also introduce a proof reconstruction mechanism based on an eauto-type algorithm combined with limited rewriting, congruence closure and some forward reasoning. The algorithm is able to re-prove in the Coq logic most of the theorems established by the ATPs. Together with machine-learning based selection of relevant premises this constitutes a full hammer system. The performance of the whole procedure is evaluated in a bootstrapping scenario emulating the development of the Coq standard library. For each theorem in the library only the previous theorems and proofs can be used. We show that 40.8% of the theorems can be proved in a push-button mode in about 40 s of real time on a 8-CPU system.

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.