  We propose a way to transform synchronous distributed algorithms solving
locally greedy and mendable problems into self-stabilizing algorithms in
anonymous networks. Mendable problems are a generalization of greedy problems
where any partial solution may be transformed -- instead of completed -- into a
global solution: every time we extend the partial solution we are allowed to
change the previous partial solution up to a given distance. Locally here means
that to extend a solution for a node, we need to look at a constant distance
from it. In order to do this, we propose the first explicit self-stabilizing
algorithm computing a $(k,k-1)$-ruling set (i.e. a "maximal independent set at
distance $k$"). By combining multiple time this technique, we compute a
distance-$K$ coloring of the graph. With this coloring we can finally simulate
\local~model algorithms running in a constant number of rounds, using the
colors as unique identifiers. Our algorithms work under the Gouda daemon, which
is similar to the probabilistic daemon: if an event should eventually happen,
it will occur under this daemon.
