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.
Other interactive theorem provers we could have based a learning environment on include Mizar @cite_10 , Isabelle @cite_12 , HOL4 @cite_34 , and Lean @cite_19 . The Mizar mathematical library is probably the most comprehensive formalization effort, but its declarative style makes it hard to employ proof search, and its source code is not freely available. Like Coq and HOL Light, also Isabelle @cite_12 was used for major formalization efforts, such as the formalization of the seL4 microkernel @cite_33 . We are not aware of a comprehensive coverage of fundamental mathematics in Isabelle, HOL4, or Lean.
@cite_33 @cite_19 @cite_34 @cite_10 @cite_12
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Numerous contributions have been made for some years to allow users to exchange formal proofs between different provers. The main propositions consist in ad hoc pointwise translations, e.g. between HOL Light and Isabelle in the Flyspeck project or uses of more or less complete certificates. We propose in this paper a methodology to combine proofs coming from different theorem provers. This methodology relies on the Dedukti logical framework as a common formalism in which proofs can be translated and combined. To relate the independently developed mathematical libraries used in proof assistants, we rely on the structuring features offered by FoCaLiZe, in particular parameterized modules and inheritance to build a formal library of transfer theorems called MathTransfer. We finally illustrate this methodology on the Sieve of Eratosthenes, which we prove correct using HOL and Coq in combination. Learning-assisted automated reasoning has recently gained popularity among the users of Isabelle HOL, HOL Light, and Mizar. In this paper, we present an add-on to the HOL4 proof assistant and an adaptation of the HOL(y)Hammer system that provides machine learning-based premise selection and automated reasoning also for HOL4. We efficiently record the HOL4 dependencies and extract features from the theorem statements, which form a basis for premise selection. HOL(y)Hammer transforms the HOL4 statements in the various TPTP-ATP proof formats, which are then processed by the ATPs. We discuss the different evaluation settings: ATPs, accessible lemmas, and premise numbers. We measure the performance of HOL(y)Hammer on the HOL4 standard library. The results are combined accordingly and compared with the HOL Light experiments, showing a comparably high quality of predictions. The system directly benefits HOL4 users by automatically finding proofs dependencies that can be reconstructed by Metis. 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]. Interactive theorem proving is tackling ever larger formalization and verification projects, and there is a critical need for theory engineering techniques to support these efforts. One such technique is cross-prover package management, which has the potential to simplify the development of logical theories and effectively share theories between different theorem prover implementations. The OpenTheory project has developed standards for packaging theories of the higher order logic implemented by the HOL family of theorem provers. What is currently missing is a standard theory library that can serve as a published contract of interoperability and contain proofs of basic properties that would otherwise appear in many theory packages. The core contribution of this paper is the presentation of a standard theory library for higher order logic represented as an OpenTheory package. We identify the core theory set of the HOL family of theorem provers, and describe the process of instrumenting the HOL Light theorem prover to extract a standardized version of its core theory development. We profile the axioms and theorems of our standard theory library and investigate the performance cost of separating the standard theory library into coherent hierarchical theory packages.