Consider a renormalizable Quantum Field Theory with scalar couplings, Yukawa couplings, and gauge coupling corresponding to a simple gauge group. The couplings \{I\}=\{\lambda, y, \bar{y}, g\} can be treated as points of a manifold in a "coupling-space", and it has been shown that there exists a function A satisfying \partial_{I} A = T_{IJ}\beta^{J} , such that G_{IJ}=1/2(T_{IJ} + T_{JI}) defines a metric on this space. If G_{IJ} is positive-definite, then A satisfies the "a-theorem", i.e. for four-dimensional QFTs there exists a function of the couplings that decreases monotonically under RG flow. Perturbatively, the existence of such an A function and knowledge of the beta functions at particular orders is enough to derive consistency conditions on the form of higher-order beta functions, providing both non-trivial checks for known functions and necessary conditions for unknown functions. This talk is based on JHEP01 (2015) 138, discussing a generalization to the gauge case of work done in Nucl. Phys. B883 (2014) 425.