An Agda formalisation of one variant of χ, along with a number of properties:
Code listing (with hyperlinked identifiers).
The code can be obtained by running the following commands (in an empty directory):
git clone http://www.cse.chalmers.se/~nad/repos/equality/
git clone http://www.cse.chalmers.se/~nad/repos/chi/
At the time of writing the code is not accepted by the most recent version of Agda (2.6.0.1), a sufficiently recent development version is required.