  The orthoscheme complex of a graded poset is a metrization of its order
complex such that the simplex of each maximal chain is isometric to the
Euclidean simplex of vertices $0, e_1,e_1+e_2,\ldots, e_1+e_2+ \cdots + e_n$.
This notion was introduced by Brady and McCammond in geometric group theory,
and has applications in discrete optimization and submodularity theory. We
address a question of what posets to yield the orthoscheme complex having
CAT(0) property. The orthoscheme complex of a modular lattice is shown to be
CAT(0), and it is conjectured that this is the case for a modular semilattice.
In this paper, we prove this conjecture affirmatively. This result implies that
a larger class of weakly modular graphs yields CAT(0) complexes.
