It is known that the complementarity problems and the variational inequality problems are reformulated equivalently as a vector equation by using the natural residual or Fischer-Burmeister function. In this short paper, we first study the global convergence of a sequential injective algorithm for weakly univalent vector equation. Then, we apply the convergence analysis to the regularized smooth...

We show that, in Martin-Löf Type Theory with a hierarchy of univalent universes, the universe Un is not an n-type. At the same time, our construction leads to an n+1-type that is not an n-type, solving one of the open problems of the Univalent Foundations Program in Princeton. A formalized version of the proof is available. Background to these Notes One of the most basic consequences of the uni...

We establish a close connection between a reversible programming language based on type isomorphisms and a formally presented univalent universe. The correspondence relates combinators witnessing type isomorphisms in the programming language to paths in the univalent universe; and combinator optimizations in the programming language to 2-paths in the univalent universe. The result suggests a si...