What is the computational content of proofs? This is one of the main topics in mathematical logic, especially proof theory, that is of relevance for computer science. The well-known foundational solutions aim at rebuilding mathematics constructively almost from scratch, and include Bishop-style constructive mathematics and Martin-Löf's intuitionistic type theory, the latter most recently in the form of the so-called homotopy or univalent type theory put forward by Voevodsky.