  We provide prophet inequality algorithms for online weighted matching in
general (non-bipartite) graphs, under two well-studied arrival models, namely
edge arrival and vertex arrival. The weight of each edge is drawn independently
from an a-priori known probability distribution. Under edge arrival, the weight
of each edge is revealed upon arrival, and the algorithm decides whether to
include it in the matching or not. Under vertex arrival, the weights of all
edges from the newly arriving vertex to all previously arrived vertices are
revealed, and the algorithm decides which of these edges, if any, to include in
the matching. To study these settings, we introduce a novel unified framework
of batched prophet inequalities that captures online settings where elements
arrive in batches; in particular it captures matching under the two
aforementioned arrival models. Our algorithms rely on the construction of
suitable online contention resolution scheme (OCRS). We first extend the
framework of OCRS to batched-OCRS, we then establish a reduction from batched
prophet inequality to batched OCRS, and finally we construct batched OCRSs with
selectable ratios of 0.337 and 0.5 for edge and vertex arrival models,
respectively. Both results improve the state of the art for the corresponding
settings. For the vertex arrival, our result is tight. Interestingly, a
pricing-based prophet inequality with comparable competitive ratios is unknown.
