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Directed topology augments the concept of a topological space with a notion of directed paths. This leads to a category of directed spaces, in which the morphisms are continuous maps respecting directed paths. Directed topology thereby enables an accurate representation of computation paths in concurrent systems that usually cannot be reversed. Even though ideas from algebraic topology have analogues in directed topology, the directedness drastically changes how spaces can be characterised. For instance, while an important homotopy invariant of a topological space is its fundamental groupoid, for directed spaces this has to be replaced by the fundamental category because directed paths are not necessarily reversible. In this paper, we present a Lean 4 formalisation of directed spaces and of a Van Kampen theorem for them, which allows the fundamental category of a directed space to be computed in terms of the fundamental categories of subspaces. Part of this formalisation is also a significant theory of directed spaces, directed homotopy theory and path coverings, which can serve as basis for future formalisations of directed topology. The formalisation in Lean can also be used in computer-assisted reasoning about the behaviour of concurrent systems that have been represented as directed spaces.

A recurring proof obligation in modern mathematics, ranging from textbook exercises to deep research problems, is to show that a given function is a morphism in some category: in analysis and topology, for example, we frequently need to prove that functions are continuous, while in group theory we are constantly concerned with homomorphisms. This paper describes a generic procedure that automatically discharges routine instances of this kind of proof obligation in an interactive theorem prover. The proof procedure has been implemented and found very useful in a mathematical case studies carried out using the ProofPower system

We present examples from a reference implementation of variants of Kleene algebras and Tarski's relation algebras in the theorem proving environment Isabelle/HOL. For Kleene algebras we show how models can be programmed, including sets of traces and paths, languages, binary relations, max-plus and min-plus algebras, matrices, formal power series. For relation algebras we discuss primarily proof automation in a comprehensive library and present an advanced formalisation example.