The design of modern proof assistants is faced with several sometimes conflicting goals, including scalability, extensibility, and soundness of proof checking. In this paper, we propose a new design for proof assistants, in an attempt to address some of these difficulties. The new design is characterized by a pervasive use of macros in representing and checking proofs, and a foundational format for theory files based on JSON. We realize these ideas in a prototype proof assistant called holpy, implemented in Python. We also demonstrate how proof automation can be extended using Python under this framework. Finally, we present a case study about a simple imperative language.
Apart from the new concepts proposed in this paper, the design of holpy is largely based on that of Isabelle @cite_2 . Many choices in the design, including the use of higher-order logic, structure of terms, syntax for inductive definitions, conversion combinators, or even the typesetting of HOL terms used in this paper, are taken from there. In addition, the author examined the design of HOL Light @cite_11 and HOL4 @cite_1 during this research.
@cite_11 @cite_1 @cite_2
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HOL Light is a new version of the HOL theorem prover. While retaining the reliability and programmability of earlier versions, it is more elegant, lightweight, powerful and automatic; it will be the basis for the Cambridge component of the HOL-2000 initiative to develop the next generation of HOL theorem provers. HOL Light is written in CAML Light, and so will run well even on small machines, e.g. PCs and Macintoshes with a few megabytes of RAM. This is in stark contrast to the resource-hungry systems which are the norm in this field, other versions of HOL included. Among the new features of this version axe a powerful simplifier, effective first order automation, simple higher-order matching and very general support for inductive and recursive definitions. HOL Light is an interactive proof assistant for classical higher-order logic, intended as a clean and simplified version of Mike Gordon's original HOL system. Theorem provers in this family use a version of ML as both the implementation and interaction language; in HOL Light's case this is Objective CAML (OCaml). Thanks to its adherence to the so-called LCF approach', the system can be extended with new inference rules without compromising soundness. While retaining this reliability and programmability from earlier HOL systems, HOL Light is distinguished by its clean and simple design and extremely small logical kernel. Despite this, it provides powerful proof tools and has been applied to some non-trivial tasks in the formalization of mathematics and industrial formal verification. We present a new scheme to translate mathematical developments from HOL Light to Coq, where they can be re-used and re-checked. By relying on a carefully chosen embedding of Higher-Order Logic into Type Theory, we try to avoid some pitfalls of inter-operation between proof systems. In particular, our translation keeps the mathematical statements intelligible. This translation has been implemented and allows the importation of the HOL Light basic library into Coq. 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.