  Using the recently proposed set of discontinuity relations we translate the
AdS/CFT Y-system to TBA integral equations and quantization conditions for a
large subset of excited states from the sl(2) sector of the $AdS_5 \times S^5$
string sigma-model. Our derivation provides an analytic proof of the fact that
the exact Bethe equations reduce to the Beisert-Staudacher equations in the
asymptotic limit. We also construct the corresponding T-system and show that in
the language of T-functions the energy formula reduces to a single term which
depends on a single T-function.
