We propose a deep learning-based approach to the problem of premise selection: selecting mathematical statements relevant for proving a given conjecture. We represent a higher-order logic formula as a graph that is invariant to variable renaming but still fully preserves syntactic and semantic information. We then embed the graph into a vector via a novel embedding method that preserves the information of edge ordering. Our approach achieves state-of-the-art results on the HolStep dataset, improving the classification accuracy from 83 to 90.3 .
@cite_38: 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.
@cite_26: 
@cite_1: We describe the superposition-based theorem prover E. E is a sound and complete prover for clausal first order logic with equality. Important properties of the prover include strong redundancy elimination criteria, the DISCOUNT loop proof procedure, a very flexible interface for specifying search control heuristics, and an efficient inference engine. We also discuss strength and weaknesses of the system.
@cite_32: 
@cite_44: We present an overview of various learning techniques used in automated theorem provers. We characterize the main problems arising in this context and classify the solutions to these problems from published approaches. We analyze the suitability of several combinations of solutions for different approaches to theorem proving and place these combinations in a spectrum ranging from provers using very specialized learning approaches to optimally adapt to a small class of proof problems, to provers that learn more general kinds of knowledge, resulting in systems that are less efficient in special cases but show improved performance for a wide range of problems. Finally, we suggest combinations of solutions for various proof philosophies.
@cite_40: Isabelle, which is available from http: isabelle.in.tum.de , is a generic framework for interactive theorem proving. The Isabelle Puremeta-logic allows the formalization of the syntax and inference rules of a broad range of object-logics following the general idea of natural deduction [32,33]. The logical core is implemented according to the well-known "LCF approach" of secure inferences as abstract datatype constructors in ML [16]; explicit proof terms are also available [8]. Isabelle Isarprovides sophisticated extra-logical infrastructure supporting structured proofs and specifications, including concepts for modular theory development. Isabelle HOLis a large application within the generic framework, with plenty of logic-specific add-on tools and a large theory library. Other notable object-logics are Isabelle ZF(Zermelo-Fraenkel set-theory, see [34,36] and Isabelle HOLCF[26] (Scott's domain theory within HOL). Users can build further formal-methods tools on top, e.g. see [53].
Research on automated theorem proving has a long history @cite_32 . Decades of research has resulted in a variety of well-developed automated theorem provers such as Coq @cite_44 , Isabelle @cite_40 , and E @cite_1 . However, no existing automated provers can scale to large mathematical libraries due to combinatorial explosion of the search space. This limitation gave rise to the development of interactive theorem proving @cite_32 , which combines humans and machines in theorem proving and has led to impressive achievements such as the proof of the Kepler conjecture @cite_38 and the formal proof of the Feit-Thompson problem @cite_26 .