  For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of
univalent universes, we show that U(n) is not an n-type. Our construction also
solves the problem of finding a type that strictly has some high truncation
level without using higher inductive types. In particular, U(n) is such a type
if we restrict it to n-types. We have fully formalized and verified our results
within the dependently typed language and proof assistant Agda.
