One appealing a strategy that relies on an infimal post-composition approach very recently developed, we provide an alternative formulation of proximal split minimization problems. Then, Forward-Backward and Douglas-Rachford splitting algorithms will guide both the design and analysis of some split numerical methods. Their globally weakly convergence and remarks on the fact that these algorithms can be equipped with relaxed and/or inertial steps leading to improved 5 convergence guarantees are given.