  Let $\R(\cdot)$ stand for the bounded-error randomized query complexity. We
show that for any relation $f \subseteq \{0,1\}^n \times \mathcal{S}$ and
partial Boolean function $g \subseteq \{0,1\}^n \times \{0,1\}$, $\R_{1/3}(f
\circ g^n) = \Omega(\R_{4/9}(f) \cdot \sqrt{\R_{1/3}(g)})$. Independently of
us, Gavinsky, Lee and Santha \cite{newcomp} proved this result. By an example
demonstrated in their work, this bound is optimal. We prove our result by
introducing a novel complexity measure called the \emph{conflict complexity} of
a partial Boolean function $g$, denoted by $\chi(g)$, which may be of
independent interest. We show that $\chi(g) = \Omega(\sqrt{\R(g)})$ and $\R(f
\circ g^n) = \Omega(\R(f) \cdot \chi(g))$.
