  We formalise, in Coq, the opening sections of Parity Complexes [Street1991]
up to and including the all important excision of extremals algorithm. Parity
complexes describe the essential combinatorial structure exhibited by
simplexes, cubes and globes, that enable the construction of free
$\omega$-categories on such objects. The excision of extremals is a recursive
algorithm that presents every cell in such a category as a (unique) composite
of atomic cells. This is the sense in which the $\omega$-category is (freely)
generated from its atoms. Due to the complicated multi-dimensional nature of
this work, the detail of definitions and proofs can be hard to follow and
verify. Indeed, some corrections were required some years following the
original publication~\cite{Street1994}. Our formalisation verifies that all
cases of each result operate as stated. In particular, we indicate which
portions of the theory can be proved directly from definitions, and which
require more subtle and complex arguments. By identifying results that require
the most complicated proofs, we are able to investigate where this theory might
benefit from further study and which results need to be considered most
carefully in future work.
