Frex: dependently-typed algebraic simplification

We present an extensible, mathematically-structured algebraic simplification library design. We structure the library using universal algebraic concepts: a free algebra -- fral -- and a free extension -- frex -- of an algebra by a set of variables. The library's dependently-typed API guarantees simplification modules, even user-defined ones, are terminating, sound, and complete with respect to a well-specified class of equations. Completeness offers intangible benefits in practice -- our main contribution is the novel design. Cleanly separating between the interface and implementation of simplification modules provides two new modularity axes. First, simplification modules share thousands of lines of infrastructure code dealing with term-representation, pretty-printing, certification, and macros/reflection. Second, new simplification modules can reuse existing ones. We demonstrate this design by developing simplification modules for monoid varieties: ordinary, commutative, and involutive. We implemented this design in the new Idris2 dependently-typed programming language, and in Agda

homotopy.io: a proof assistant for finitely-presented globular nn-categories

We present the proof assistant homotopy.io for working with finitely-presented semistrict higher categories. The tool runs in the browser with a point-and-click interface, allowing direct manipulation of proof objects via a graphical representation. We describe the user interface and explain how the tool can be used in practice. We also describe the essential subsystems of the tool, including collapse, contraction, expansion, typechecking, and layout, as well as key implementation details including data structure encoding, memoisation, and rendering. These technical innovations have been essential for achieving good performance in a resource-constrained setting