Recent works on bounding the output size of a conjunctive query with functional dependencies and degree constraints have shown a deep connection between fundamental questions in information theory and database theory. We prove analogous output bounds for disjunctive datalog rules, and answer several open questions regarding the tightness and looseness of these bounds along the way. Our bounds are intimately related to Shannon-type information inequalities. We devise the notion of a "proof sequence" of a specific class of Shannon-type information inequalities called "Shannon flow inequalities". We then show how such a proof sequence can be interpreted as symbolic instructions guiding an algorithm called "PANDA", which answers disjunctive datalog rules within the time that the size bound predicted. We show that PANDA can be used as a black-box to devise algorithms matching precisely the fractional hypertree width and the submodular width runtimes for aggregate and conjunctive queries with functional dependencies and degree constraints. Our results improve upon known results in three ways. First, our bounds and algorithms are for the much more general class of disjunctive datalog rules, of which conjunctive queries are a special case. Second, the runtime of PANDA matches precisely the submodular width bound, while the previous algorithm by Marx has a runtime that is polynomial in this bound. Third, our bounds and algorithms work for queries with input cardinality bounds, functional dependencies, and degree constraints. Overall, our results show a deep connection between three seemingly unrelated lines of research; and, our results on proof sequences for Shannon flow inequalities might be of independent interest.