\section{Computer checks}
\label{rw:app:computer}

Some of the computations in this paper were also carried out by
computer. No proof depends on such a computation: in each case the
argument is given in the text, and the program is a check on it.
Statements about a classical group of
arbitrary rank can be checked by a program only in the ranks
indicated, and for the other ranks the argument in the text is the
only proof.

The programs are written in Python, with the libraries SymPy and
NumPy, and are in the directory \texttt{anc} of the source of this
paper. The program
\texttt{exchange\_checker.py} is a common
library. It keeps track of a set of roots and of the roots on
which a character is not trivial, and at each exchange it tests
that the sets are closed and normalized under addition of roots,
that the character has the required support, and that the sum of
the two exchanged roots lies in this support. These are the
conditions on roots in Lemma~\ref{fl:lem:odd-exchange}; the
constants of the commutators and the Fourier pairing over the
adeles are treated in the text. The other programs compute with
matrices, for subspaces of Lie algebras or for the groups
themselves.

\begingroup
\small
\renewcommand{\arraystretch}{1.15}
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{.25\textwidth}
                    >{\raggedright\arraybackslash}p{.31\textwidth}
                    >{\raggedright\arraybackslash}p{.38\textwidth}@{}}
\toprule
Statement & What is checked & Program and range\\
\midrule
\endfirsthead
\toprule
Statement & What is checked & Program and range\\
\midrule
\endhead
\bottomrule
\endfoot
Lemmas~\ref{rw:lem:jacquet-double-coset}
and~\ref{rw:lem:jacquet-intersection} &
exactly one double coset of Weyl elements contributes, and the
intersection of unipotent groups is as stated &
\texttt{\footnotesize check\_orthogonal\_jacquet.py}; $B_2,B_4,B_6$ (also with
the criterion of characteristic two), $D_4,D_6$\\
Lemma~\ref{fl:lem:uniform-interleave} &
for the two chains, the conditions on the sets of roots and on
the supports of the characters &
\texttt{\footnotesize check\_interleave.py}, $2\le r\le7$\\
Proposition~\ref{fl:prop:uniform-b-nonzero}, steps~1, 2 and~4 &
the sequences of exchanges and the final subgroups and characters &
\texttt{\footnotesize check\_odd\_descent\_steps12.py}, $n\le4$;
\texttt{\footnotesize check\_sw\_exchange.py}, $m\le12$\\
Proposition~\ref{fl:prop:uniform-b-nonzero}, step~3 &
the hyperbolic basis, the determinant of $P_t$ and the conjugation
formula &
\texttt{\footnotesize check\_odd\_descent.py}; $n\le5$\\
Lemma~\ref{rw:lem:jacobi-embedding} &
the embedding of the Jacobi group and the commutator used in
Lemma~\ref{fl:lem:descent-vanishing-and-period} &
\texttt{\footnotesize check\_jacobi\_embedding.py}; symbolic\\
Lemma~\ref{fl:lem:even-period-to-generic}, step~2 &
the equivalence of the two Fourier coefficients, in the Lie
algebra, and the additivity of $\Lambda_c$ on the group &
\texttt{\footnotesize check\_even\_descent\_wh.py}, $n=2,3,4$,
with several rational values of $c$;
\texttt{\footnotesize check\_lambda\_c.py}, $n=2$\\
Lemma~\ref{fl:lem:descent-vanishing-and-period}, steps~2 and~3 &
the chain of exchanges and the signs in the final table &
\texttt{\footnotesize check\_even\_descent\_chain.py}; $n\le4$\\
\eqref{fl:eq:char2-root-commutator},
\eqref{rw:eq:terminal-commutator},
\eqref{rw:eq:terminal-conjugation},
Corollary~\ref{fl:cor:char2-descent-split} &
matrix identities over polynomial rings in characteristic two &
\texttt{\footnotesize check\_char2.py}; symbolic\\
Lemma~\ref{fl:lem:char2-row-iteration}, step~5;
Proposition~\ref{fl:prop:char2-direct-generic}, steps~1--3 &
the sets of roots &
\texttt{\footnotesize check\_roots.py}; $m\le9$, $n\le7$\\
\end{longtable}
\endgroup