The Common HOL Platform

The Common HOL project aims to facilitate porting source code and proofs between members of the HOL family of theorem provers. At the heart of the project is the Common HOL Platform, which defines a standard HOL theory and API that aims to be compatible with all HOL systems. So far, HOL Light and hol90 have been adapted for conformance, and HOL Zero was originally developed to conform. In this paper we provide motivation for a platform, give an overview of the Common HOL Platform's theory and API components, and show how to adapt legacy systems. We also report on the platform's successful application in the hand-translation of a few thousand lines of source code from HOL Light to HOL Zero.Comment: In Proceedings PxTP 2015, arXiv:1507.0837

Conversion of HOL Light proofs into Metamath

We present an algorithm for converting proofs from the OpenTheory interchange format, which can be translated to and from any of the HOL family of proof languages (HOL4, HOL Light, ProofPower, and Isabelle), into the ZFC-based Metamath language. This task is divided into two steps: the translation of an OpenTheory proof into a Metamath HOL formalization, $\mathtt{\text{hol.mm}}$, followed by the embedding of the HOL formalization into the main ZFC foundations of the main Metamath library, $\mathtt{\text{set.mm}}$. This process provides a means to link the simplicity of the Metamath foundations to the intense automation efforts which have borne fruit in HOL Light, allowing the production of complete Metamath proofs of theorems in HOL Light, while also proving that HOL Light is consistent, relative to Metamath's ZFC axiomatization.Comment: 14 pages, 2 figures, accepted to Journal of Formalized Reasonin

From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions

Permissive-Nominal Logic (PNL) extends first-order predicate logic with term-formers that can bind names in their arguments. It takes a semantics in (permissive-)nominal sets. In PNL, the forall-quantifier or lambda-binder are just term-formers satisfying axioms, and their denotation is functions on nominal atoms-abstraction. Then we have higher-order logic (HOL) and its models in ordinary (i.e. Zermelo-Fraenkel) sets; the denotation of forall or lambda is functions on full or partial function spaces. This raises the following question: how are these two models of binding connected? What translation is possible between PNL and HOL, and between nominal sets and functions? We exhibit a translation of PNL into HOL, and from models of PNL to certain models of HOL. It is natural, but also partial: we translate a restricted subsystem of full PNL to HOL. The extra part which does not translate is the symmetry properties of nominal sets with respect to permutations. To use a little nominal jargon: we can translate names and binding, but not their nominal equivariance properties. This seems reasonable since HOL---and ordinary sets---are not equivariant. Thus viewed through this translation, PNL and HOL and their models do different things, but they enjoy non-trivial and rich subsystems which are isomorphic

Stateless HOL

We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far has been moved out of the kernel. This means that the kernel now is purely functional. The changes to the system are small. All existing HOL Light developments can be run by the stateless system with only minor changes. The basic principle behind the system is not to name constants by strings, but by pairs consisting of a string and a definition. This means that the data structures for the terms are all merged into one big graph. OCaml - the implementation language of the system - can use pointer equality to establish equality of data structures fast. This allows the system to run at acceptable speeds. Our system runs at about 85% of the speed of the stateful version of HOL Light.Comment: In Proceedings TYPES 2009, arXiv:1103.311

HOL(y)Hammer: Online ATP Service for HOL Light

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

Duality in Segal-Bargmann Spaces

For $\alpha>0$, the Bargmann projection $P_\alpha$ is the orthogonal projection from $L^2(\gamma_\alpha)$ onto the holomorphic subspace $L^2_{hol}(\gamma_\alpha)$, where $\gamma_\alpha$ is the standard Gaussian probability measure on \C^n with variance $(2\alpha)^{-n}$. The space $L^2_{hol}(\gamma_\alpha)$ is classically known as the Segal-Bargmann space. We show that $P_\alpha$ extends to a bounded operator on $L^p(\gamma_{\alpha p/2})$, and calculate the exact norm of this scaled $L^p$ Bargmann projection. We use this to show that the dual space of the $L^p$-Segal-Bargmann space $L^p_{hol}(\gamma_{\alpha p/2})$ is an $L^{p'}$ Segal-Bargmann space, but with the Gaussian measure scaled differently: $(L^p_{hol}(\gamma_{\alpha p/2}))^* \cong L^{p'}_{hol}(\gamma_{\alpha p'/2})$ (this was shown originally by Janson, Peetre, and Rochberg). We show that the Bargmann projection controls this dual isomorphism, and gives a dimension-independent estimate on one of the two constants of equivalence of the norms.Comment: 24 page

Compact Riemannian manifolds with exceptional holonomy

Suppose that M is an orientable n-dimensional manifold, and g a Riemannian metric on M. Then the holonomy group Hol(g) of g is an important invariant of g. It is a subgroup of SO(n). For generic metrics g on M the holonomy group Hol(g) is SO(n), but for some special g the holonomy group may be a proper Lie subgroup of SO(n). When this happens the metric g is compatible with some extra geometric structure on M, such as a complex structure. The possibilities for Hol(g) were classified in 1955 by Berger. Under conditions on M and g given in §1, Berger found that Hol(g) must be one of SO(n), U(m), SU(m), Sp(m), Sp(m)Sp(1), G2 or Spin(7). His methods showed that Hol(g) is intimately related to the Riemann curvature R of g. One consequence of this is that metrics with holonomy Sp(m)Sp(1) for m> 1 are automatically Einstein, and metrics with holonomy SU(m), Sp(m), G2 or Spin(7) are Ricci-flat. Now, people have found many different ways of producing examples of metrics with these holonomy groups, by exploiting the extra geometric structure – for exam-ple, quotient constructions, twistor geometry, homogeneous and cohomogeneity on