We make the case that the foundation for Rely-Guarantee reasoning can be fruitfully delivered by a coinductive semantics. Using insight from an Isabelle formalization, via a proof analysis we show that the coinductive semantics tends to simplify the proof development; in particular it enables more direct proofs for the soundness of the Rely-Guarantee rules. The comparison between inductive and coinductive proofs also suggests inductive counterparts of coinductive “up-to” enhancements. On the way, we fill a gap in the literature, by showing that three previously defined inductive semantics for Rely-Guarantee are equivalent. Underlying our transformation of an inductive into a coinductive semantics is the notion of inductively approximating a coinductive predicate—which, deployed in the opposite direction (from coinduction to induction), is a standard technical tool for approximating process algebra bisimilarities. On the spectrum between the abstract fixpoint theorems and concrete instances, we formalize effective format-based criteria that enable sound approximation.

Abstract Meltdown and Spectre are vulnerabilities known as transient execution vulnerabilities, where an attacker exploits speculative execution (a semantic optimization present in most modern processors) to break confidentiality. We introduce relative security , a general notion of information-flow security that models this type of vulnerability by contrasting the leaks that are possible in a “vanilla” semantics with those possible in a different semantics, often obtained from the vanilla semantics via some optimizations. We describe incremental proof methods, in the style of Goguen and Meseguer’s unwinding, both for proving and for disproving relative security, and deploy these to formally establish the relative (in)security of some standard Spectre examples. Both the abstract results and the case studies have been mechanized in the Isabelle/HOL theorem prover. This paper is an extension of an earlier conference paper that provides significantly more detail on the Isabelle formalization and the unwinding proof process.

Formalised libraries of combinatorial mathematics have rapidly expanded over the last five years, but few use one of the most important tools: probability. How can often intuitive probabilistic arguments on the existence of combinatorial structures, such as hypergraphs, be translated into a formal text? We present a modular framework using locales in Isabelle/HOL to formalise such probabilistic proofs, including the basic existence method and first formalisation of the Lovász local lemma, a fundamental result in probability. The formalisation focuses on general, reusable formal probabilistic lemmas for combinatorial structures, and highlights several notable gaps in typical intuitive probabilistic reasoning on paper. The applicability of the techniques is demonstrated through the formalisation of several classic lemmas on the existence of hypergraphs with certain colourings.

We describe our formalisation in the interactive theorem prover Isabelle/HOL of the Balog–Szemerédi–Gowers Theorem, a profound result in additive combinatorics which played a central role in Gowers’s proof deriving the first effective bounds for Szemerédi’s Theorem. The proof is of great mathematical interest given that it involves an interplay between different mathematical areas, namely applications of graph theory and probability theory to additive combinatorics involving algebraic objects. This interplay is what made the process of the formalisation, for which we had to develop formalisations of new background material in the aforementioned areas, more rich and technically challenging. We demonstrate how locales, Isabelle’s module system, can be employed to handle such interplays in mathematical formalisations. To treat the graph-theoretic aspects of the proof, we make use of a new, more general undirected graph theory library developed by Edmonds, which is both flexible and extensible. In addition to the main theorem, which, following our source, is formulated for difference sets, we also give an alternative version for sumsets which required a formalisation of an auxiliary triangle inequality. We moreover formalise a few additional results in additive combinatorics that are not used in the proof of the main theorem. This is the first formalisation of the Balog–Szemerédi–Gowers Theorem in any proof assistant to our knowledge.

Abstract We have formalised Szemerédi’s Regularity Lemma and Roth’s Theorem on Arithmetic Progressions, two major results in extremal graph theory and additive combinatorics, using the proof assistant Isabelle/HOL. For the latter formalisation, we used the former to first show the Triangle Counting Lemma and the Triangle Removal Lemma: themselves important technical results. Here, in addition to showcasing the main formalised statements and definitions, we focus on sensitive points in the proofs, describing how we overcame the difficulties that we encountered.

The formalisation of mathematics is continuing rapidly, however combinatorics continues to present challenges to formalisation efforts, such as its reliance on techniques from a wide range of other fields in mathematics. This paper presents formal linear algebraic techniques for proofs on incidence structures in Isabelle/HOL, and their application to the first formalisation of Fisher's inequality. In addition to formalising incidence matrices and simple techniques for reasoning on linear algebraic representations, the formalisation focuses on the linear algebra bound and rank arguments. These techniques can easily be adapted for future formalisations in combinatorics, as we demonstrate through further application to proofs of variations on Fisher's inequality.