  This paper studies variable-length (VL) source coding of general sources with
side-information. Novel one-shot coding theorems for coding with common
side-information available at the encoder and the decoder and Slepian- Wolf
(SW) coding (i.e., with side-information only at the decoder) are given, and
then, are applied to asymptotic analyses of these coding problems. Especially,
a general formula for the infimum of the coding rate asymptotically achievable
by weak VL-SW coding (i.e., VL-SW coding with vanishing error probability) is
derived. Further, the general formula is applied to investigating weak VL-SW
coding of mixed sources. Our results derive and extend several known results on
SW coding and weak VL coding, e.g., the optimal achievable rate of VL-SW coding
for mixture of i.i.d. sources is given for countably infinite alphabet case
with mild condition. In addition, the usefulness of the encoder
side-information is investigated. Our result shows that if the encoder
side-information is useless in weak VL coding then it is also useless even in
the case where the error probability may be positive asymptotically.
