Sharing a Library between Proof Assistants: Reaching out to the HOL Family

Abstract

We observe today a large diversity of proof systems. This diversity has the negative consequence that a lot of theorems are proved many times. Unlike programming languages, it is difficult for these systems to co-operate because they do not implement the same logic. Logical frameworks are a class of theorem provers that overcome this issue by their capacity of implementing various logics. In this work, we study the STT∀ βδ logic, an extension of Simple Type Theory that has been encoded in the logical framework Dedukti [6]. We present a translation from this logic to OpenTheory [8], a proof system and interoperability tool between provers of the HOL family. We have used this translation to export an arithmetic library containing Fermat's little theorem to OpenTheory and to two other proof systems that are Coq [13] and Matita [2].

Safe, fast, concurrent proof checking for the lambda-pi calculus modulo rewriting

“…Among the ATP datasets, I evaluate proofs of TPTP problems generated by iProver Modulo and proofs of theorems from B method set theory generated by Zenon Modulo [9]. For the ITP datasets, I evaluate parts of the standard libraries from HOL Light (up to finite Cartesian products) and Isabelle/HOL (up to HOL.List), as well as Fermat's little theorem proved in Matita [31]. An evaluation of Coq datasets is unfortunately not possible because its encoding relies on higher-order rewriting.…”

Safe, Fast, Concurrent Proof Checking for the lambda-Pi Calculus Modulo Rewriting

“…Among the ATP datasets, I evaluate proofs of TPTP problems generated by iProver Modulo and proofs of theorems from B method set theory generated by Zenon Modulo [9]. For the ITP datasets, I evaluate parts of the standard libraries from HOL Light (up to finite Cartesian products) and Isabelle/HOL (up to HOL.List), as well as Fermat's little theorem proved in Matita [34]. An evaluation of Coq datasets is unfortunately not possible due to its current encoding relying on higher-order rewriting.…”

Experiences from Exporting Major Proof Assistant Libraries

“…But these approaches lived only in the logical framework and lacked a connection to the actual systems and their libraries. Recently, small arithmetic libraries were transported in this way using Dedukti as an intermediate [61].…”

“…Even though the alignment problem was recognized in [50] already, which also gives the first solution in a practical setting, it has remained difficult. Major case studies so far are [61] for aligning many different small libraries and [18] for aligning two large ones. We gave a survey of issues in [28], and if anything, we see a longer list of difficulties by now.…”