The nuclear Energy Density Functional (EDF) approach is used to study medium-mass and heavy nuclei in a systematic manner [1]. Even though currently used EDFs provide a satisfactory description of low-energy properties of known nuclei, their empirical character and the spreading of the results obtained from different parameterizations as one moves away from the valley of !-stability and enters experimentally-unexplored regions point to the lack of predictive power of today's calculations. Our objective is to improve on such a situation by designing non-empirical energy density functionals constrained explicitly from inter-nucleon interactions in the vacuum. As a starting point, we have performed the first systematic finite-nuclei calculations using a nuclear EDF whose pairing part is derived from low-momentum [2] two-nucleon interactions in the vacuum. At present, calculations have been performed for all semi-magic nuclei employing a pairing functional derived at lowest-order in the nuclear plus Coulomb two-nucleon interaction [3,4]. The analysis of the results and of their comparison with existing experimental data allow us to outline three important points. (i) The Coulomb interaction has a significant impact on proton-proton superfluidity in nuclei. (ii) Lowest-order calculations lead to qualitatively different results depending on whether one starts from a high-cutoff nuclear Hamiltonian or from a low-cut-off one [5]. (iii) Using a low-momentum nuclear Hamiltonian, as is recommended here, the agreement between theoretical and experimental pairing gaps put stringent constraints on the overall contribution from missing ingredients: partial waves with L > 0, the three-nucleon interaction and higher-order effects, e.g. the coupling to density/spin/isospin fluctuations. In order to reduce the computational cost of such non-empirical calculations and perform systematic symmetry-unrestricted calculations, is it of interest to design empirical local pairing functionals that reproduce the results provided by non-empirical ones. Taking our lowest-order results as an intermediate reference, we investigate the needed isoscalar- and isovector-density dependencies [6] of the empirical local pairing functional to do so. In this modeling, we explicitly separate the part of the pairing functional accounting for the Coulomb anti proton-pairing effect.