Building a summary for library code is a common approach to speeding up the analysis of client code. In presence of callbacks, some reachability relationships between library nodes cannot be obtained during library-code summarization. Thus, the library code may have to be analyzed again during the analysis of the client code with the library summary. In this paper, we propose to summarize library code with tree-adjoining-language (TAL) reachability. Compared with the summary built with context-free-language (CFL) reachability, the summary built with TAL reachability further contains conditional reachability relationships. The conditional reachability relationships can lead to much lighter analysis of the library code during the client code analysis with the TAL-reachability-based library summary. We also performed an experimental comparison of context-sensitive data-dependence analysis with the TAL-reachability-based library summary and context-sensitive data-dependence analysis with the CFL-reachability-based library summary using 15 benchmark subjects. Our experimental results demonstrate that the former has an 8X speed-up over the latter on average.