An equivalence relation can be constructed from a given (homogeneous, binary) relation in two steps: first, construct the smallest reflexive and transitive relation containing the given relation (the “star” of the relation) and, second, construct the largest symmetric relation that is included in the result of the first step. The fact that the final result is also reflexive and transitive (as well as symmetric), and thus an equivalence relation, is not immediately obvious, although straightforward to prove. Rather than prove that the defining properties of reflexivity and transitivity are satisfied, we establish reflexivity and transitivity constructively by exhibiting a starth root—in a way that emphasises the creative process in its construction. The resulting construction is fundamental to algorithms that determine the strongly connected components of a graph as well as the decomposition of a graph into its strongly connected components together with an acyclic graph connecting such components.

This theoretical pearl shows how a graphical, relational, point-free, and calculational approach to linear algebra, known as graphical linear algebra, can be used to reason not only about matrices (and matrix algebra, as can be found in the literature) but also vector spaces and more generally linear relations. Linear algebra is usually seen as the study of vector spaces and linear transformations. However, to reason effectively with subspaces in a point-free and calculational manner, both can be generalized to an unifying concept: linear relations, much like relational algebra. While the semantics is relational, the syntax is graphical and uses string diagrams, 2-dimensional formal diagrams, which represent the linear relations. Most importantly, in a number of cases, the relational semantics allows algorithms and properties to be derived calculationally instead of just verified. Our approach is to proceed primarily by examples which involve finding inverses, switching from an implicit basis to an explicit basis (solving a homogeneous linear system), exploring both the exchange lemma and the Zassenhaus’ algorithm.

Consider two widely used definitions of equality. That of Leibniz: one value equals another if any predicate that holds of the first holds of the second. And that of Martin-Löf: the type identifying one value with another is occupied if the two values are identical. The former dates back several centuries, while the latter is widely used in proof systems such as Agda and Coq. Here we show that the two definitions are isomorphic: we can convert any proof of Leibniz equality to one of Martin-Löf identity and vice versa, and each conversion followed by the other is the identity. One direction of the isomorphism depends crucially on values of the type corresponding to Leibniz equality satisfying functional extensionality and Reynolds’ notion of parametricity. The existence of the conversions is widely known (meaning that if one can prove one equality then one can prove the other), but that the two conversions form an isomorphism (internally) in the presence of parametricity and functional extensionality is, we believe, new. Our result is a special case of a more general relation that holds between inductive families and their Church encodings. Our proofs are given inside type theory, rather than meta-theoretically. Our paper is a literate Agda script.