  In homotopy type theory, the truncation operator ||-||n (for a number n > -2)
is often useful if one does not care about the higher structure of a type and
wants to avoid coherence problems. However, its elimination principle only
allows to eliminate into n-types, which makes it hard to construct functions
||A||n -> B if B is not an n-type. This makes it desirable to derive more
powerful elimination theorems. We show a first general result: If B is an
(n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B
which are constant on all (n+1)-st loop spaces. We give one "elementary" proof
and one proof that uses a higher inductive type, both of which require some
effort. As a sample application of our result, we show that we can construct
"set-based" representations of 1-types, as long as they have "braided" loop
spaces. The main result with one of its proofs and the application have been
formalised in Agda.
