Suppose one has found a collection of Lagrangians in a Calabi-Yau manifold whose Fukaya algebra A is non-vanishing and homologically smooth in the sense of non-commutative geometry, a condition intrinsic to A. Then, we show the collection automatically split-generates the Fukaya category. In addition, the Hochschild invariants of the algebra (and hence of the whole Fukaya category) are automatically isomorphic to the quantum cohomology ring of the manifold. If the target manifold was non-Calabi-Yau (e.g., Fano), the same result holds, under an additional hypothesis on A being cohomologically large enough. The proofs make large use of joint work with T. Perutz and N. Sheridan, which in turn is part of a further story about recovering Gromov-Witten invariants from the Fukaya category, and enumerative mirror symmetry from homological mirror symmetry.