  In this paper we present and illustrate a general methodology to apply KAM
theory in particular problems, based on an {\em a posteriori} approach. We
focus on the existence of real-analytic quasi-periodic Lagrangian invariant
tori for symplectic maps. The purpose is to verify the hypotheses of a KAM
theorem in an a posteriori format: given a parameterization of an approximately
invariant torus, we have to check non-resonance (Diophantine) conditions,
non-degeneracy conditions and certain inequalities to hold. To check such
inequalities we require to control the analytic norm of some functions that
depend on the map, the ambient structure and the parameterization. To this end,
we propose an efficient computer assisted methodology, using fast Fourier
transform, having the same asymptotic cost of using the parameterization method
for obtaining numerical approximations of invariant tori. We illustrate our
methodology by proving the existence of invariant curves for the standard map
(up to $\eps=0.9716$), meandering curves for the non-twist standard map and
2-dimensional tori for the Froeschl\'e map.
