You have proof that you should push the fat man?

Lengthy breakdown of my response.

TL;DR: You should push the fat man if and only if X. You should not push the fat man if and only if ¬X.

X can be derived into a rule to use with D(X') to compute whether you should push or not. X can also be derived into a utility function to use with U(X') to compute whether you should push or not. The answer in either case doesn't depend on U or D, it depends on your derivation of X, which itself depends on X.

This is shown by the assumption that for all reasonable a, there exists a g(a) where U(a) = D(g(a)). Since, by their ambiguity and vague definitions, both U() and D() seem to cover an infinite domain and are the equivalent of turing-complete, this assumption seems very natural.