Sterling: Proven Normalization for Cubical Type Theory 🔍

We prove normalization for univalent cubical type theory using Artin gluing, obtaining two important syntactic results as corollaries: decidability of judgmental equality and injectivity of type constructors. Our result is carried out in the internal language of a glued topos, which we refer to as “synthetic Tait computability” by analogy with synthetic differential geometry and synthetic domain theory.