text: We present an environment, benchmark, and deep learning driven automated theorem prover for higher-order logic. Higher-order interactive theorem provers enable the formalization of arbitrary mathematical theories and thereby present an interesting, open-ended challenge for deep learning. We provide an open-source framework based on the HOL Light theorem prover that can be used as a reinforcement learning environment. HOL Light comes with a broad coverage of basic mathematical theorems on calculus and the formal proof of the Kepler conjecture, from which we derive a challenging benchmark for automated reasoning. We also present a deep reinforcement learning driven automated theorem prover, DeepHOL, with strong initial results on this benchmark.
  @cite_18: In this paper, we introduce a system called GamePad that can be used to explore the application of machine learning methods to theorem proving in the Coq proof assistant. Interactive theorem provers such as Coq enable users to construct machine-checkable proofs in a step-by-step manner. Hence, they provide an opportunity to explore theorem proving with human supervision. We use GamePad to synthesize proofs for a simple algebraic rewrite problem and train baseline models for a formalization of the Feit-Thompson theorem. We address position evaluation (i.e., predict the number of proof steps left) and tactic prediction (i.e., predict the next proof step) tasks, which arise naturally in tactic-based theorem proving.
  @cite_14: A practical introduction to the development of proofs and certified programs using Coq. An invaluable tool for researchers, students, and engineers interested in formal methods and the development of zero-fault software.
summary: GamePad has very similar objectives to ours @cite_18 . They also provide an easy-to-use Python API for an interactive theorem prover, and they present test and training sets. They chose to base their system on Coq @cite_14 , an interactive theorem prover based on the calculus of inductive constructions. While enabling automatic code extraction, it comes with a much smaller coverage of fundamental mathematics. Even including the formalization of the Feit-Thompson theorem, their benchmark comprises only 1602 theorems and lemmas, while ours features 29462 theorems and lemmas. Besides presenting a much larger data set, we also demonstrate the feasibility of achieving state-of-the-art prover performance based on our data and environment by presenting a deep learning based theorem prover. We also report the results as theorem proving performance instead of proxy metrics.