  Several types of dependencies have been proposed for the static analysis of
existential rule ontologies, promising insights about computational properties
and possible practical uses of a given set of rules, e.g., in ontology-based
query answering. Unfortunately, these dependencies are rarely implemented, so
their potential is hardly realised in practice. We focus on two kinds of rule
dependencies -- positive reliances and restraints -- and design and implement
optimised algorithms for their efficient computation. Experiments on real-world
ontologies of up to more than 100,000 rules show the scalability of our
approach, which lets us realise several previously proposed applications as
practical case studies. In particular, we can analyse to what extent rule-based
bottom-up approaches of reasoning can be guaranteed to yield redundancy-free
"lean" knowledge graphs (so-called cores) on practical ontologies.
