This is a very sophisticated problem, involving rewrite theory. Without the compositions, and with only one connection, there is work in 86. (with A.P.TONKS), ``Calculations with simplicial and cubical groups in AXIOM'', {\em J. Symbolic Computation}, 17 (1994) 159-179. But AXIOM is not the easiest to get working for you! I have not looked at this for a long time. The difficulties of using the interchange rule are quickly seen by considering pre crossed and crossed modules. We actually did some calculations of induced crossed modules in 92. (with C.D.WENSLEY), ``On finite induced crossed modules and the homotopy 2-type of mapping cones'', {\em Theory and Applications of Categories} 1(3) (1995) 54-71. 95. (with C.D.WENSLEY), ``Computing crossed modules induced by an inclusion of a normal subgroup, with applications to homotopy 2-types'', {\em Theory and Applications of Categories} 2 (1996) 3-16. There is more on rewriting in Heyworth-Wensley Mathematics, abstract math.CO/9907082 Logged Rewriting Procedures with Application to Identities Among Relations I thoroughly appreciate the need! Ronnie Brown On Thu, 15 Jul 1999, Philippe Gaucher wrote:
Bonjour,
I have calculations to do in a cubical omega-category (see works of Brown, Higgins, etc... for the definition). I have to verify some equalities. I am wondering whether there is a program (Maple, anything else) in order to simplify automatically expressions containing only the usual operators like the three families of degeneracy maps of a cubical set with connections, operations +_j, and the usual face maps.
Every composition of degeneracy maps and face maps can be ordered with the degeneracy maps of the cubical sets in the first place (in a canonical order), followed by the connection maps, followed by the face maps. But I do not see a canonical way to deal with +_j (because of the interchange law for example).
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