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Formally reasoning about functional programs is supposed to be straightforward and elegant, however, it is not typically done as a matter of course. Reasoning in a proof assistant requires "reimplementing" the code in those tools, which is far from trivial. SMLtoCoq provides an automatic translation of SML programs and function contracts into Coq. Programs are translated into Coq specifications, and function contracts into theorems, which can then be formally proved. Using the Equations plugin and other well established Coq libraries, SMLtoCoq is able to translate SML programs without side-effects containing partial functions, structures, functors, records, among others. Additionally, we provide a Coq version of many parts of SML's basis library, so that calls to these libraries are kept almost as is., Comment: In Proceedings LFMTP 2021, arXiv:2107.07376

Structural proof theory is praised for being a symbolic approach to reasoning and proofs, in which one can define schemas for reasoning steps and manipulate proofs as a mathematical structure. For this to be possible, proof systems must be designed as a set of rules such that proofs using those rules are correct by construction. Therefore, one must consider all ways these rules can interact and prove that they satisfy certain properties which makes them "well-behaved". This is called the meta-theory of a proof system. Meta-theory proofs typically involve many cases on structures with lots of symbols. The majority of cases are usually quite similar, and when a proof fails, it might be because of a sub-case on a very specific configuration of rules. Developing these proofs by hand is tedious and error-prone, and their combinatorial nature suggests they could be automated. There are various approaches on how to automate, either partially or completely, meta-theory proofs. In this paper, I will present some techniques that I have been involved in for facilitating meta-theory reasoning., Comment: In Proceedings LFMTP 2021, arXiv:2107.07376

This volume of EPTCS contains the proceedings of the Sixth Workshop on Proof Exchange for Theorem Proving (PxTP 2019), held on 26 August 2019 as part of the CADE-27 conference in Natal, Brazil. The PxTP workshop series brings together researchers working on various aspects of communication, integration, and cooperation between reasoning systems and formalisms, with a special focus on proofs. The progress in computer-aided reasoning, both automated and interactive, during the past decades, made it possible to build deduction tools that are increasingly more applicable to a wider range of problems and are able to tackle larger problems progressively faster. In recent years, cooperation between such tools in larger systems has demonstrated the potential to reduce the amount of manual intervention. Cooperation between reasoning systems relies on availability of theoretical formalisms and practical tools to exchange problems, proofs, and models. The PxTP workshop series strives to encourage such cooperation by inviting contributions on all aspects of cooperation between reasoning tools, whether automatic or interactive.

Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's intuitionistic theorems in the traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems., Comment: In Proceedings Linearity-TLLA 2018, arXiv:1904.06159

We present a declarative and modular specification of an automated trading system (ATS) in the concurrent linear framework CLF. We implemented it in Celf, a CLF type checker which also supports executing CLF specifications. We outline the verification of two representative properties of trading systems using generative grammars, an approach to reasoning about CLF specifications., Comment: In Proceedings Linearity-TLLA 2018, arXiv:1904.06159

This volume contains a selection of papers presented at LFMTP 2018, the 13th international Workshop on Logical Frameworks and Meta-Languages: Theory and Practice (LFMTP), held on July 7, 2018, in Oxford, UK. The workshop was affiliated with the 3rd international conference on Formal Structures for Computation and Deduction (FSCD) within the 7th Federated Logic Conference (FLoC).

This paper tackles the problem of formulating and proving the completeness of focused-like proof systems in an automated fashion. Focusing is a discipline on proofs which structures them into phases in order to reduce proof search non-determinism. We demonstrate that it is possible to construct a complete focused proof system from a given un-focused proof system if it satisfies some conditions. Our key idea is to generalize the completeness proof based on permutation lemmas given by Miller and Saurin for the focused linear logic proof system. This is done by building a graph from the rule permutation relation of a proof system, called permutation graph. We then show that from the permutation graph of a given proof system, it is possible to construct a complete focused proof system, and additionally infer for which formulas contraction is admissible. An implementation for building the permutation graph of a system is provided. We apply our technique to generate the focused proof systems MALLF, LJF and LKF for linear, intuitionistic and classical logics, respectively., Comment: In Proceedings WoF'15, arXiv:1511.02529

Different automated theorem provers reason in various deductive systems and, thus, produce proof objects which are in general not compatible. To understand and analyze these objects, one needs to study the corresponding proof theory, and then study the language used to represent proofs, on a prover by prover basis. In this work we present an implementation that takes SMT and Connection proof objects from two different provers and imports them both as expansion trees. By representing the proofs in the same framework, all the algorithms and tools available for expansion trees (compression, visualization, sequent calculus proof construction, proof checking, etc.) can be employed uniformly. The expansion proofs can also be used as a validation tool for the proof objects produced., Comment: In Proceedings PxTP 2015, arXiv:1507.08375

Cut-introduction is a technique for structuring and compressing formal proofs. In this paper we generalize our cut-introduction method for the introduction of quantified lemmas of the form $\forall x.A$ (for quantifier-free $A$) to a method generating lemmas of the form $\forall x_1\ldots\forall x_n.A$. Moreover, we extend the original method to predicate logic with equality. The new method was implemented and applied to the TSTP proof database. It is shown that the extension of the method to handle equality and quantifier-blocks leads to a substantial improvement of the old algorithm.

We describe a method for inverting Gentzen's cut-elimination in classical first-order logic. Our algorithm is based on first computign a compressed representation of the terms present in the cut-free proof and then cut-formulas that realize such a compression. Finally, a proof using these cut-formulas is constructed. This method allows an exponential compression of proof length. It can be applied to the output of automated theorem provers, which typically produce analytic proofs. An implementation is available on the web and described in this paper.