Decreasing Diagrams are Complete for Confluence
Confluence is a fundamental property of nondeterministic computations, arising from parallelism, concurrency, or freedom in the evaluation order. It guarantees that such a computation always yields the same result, regardless of the order in which steps are taken. The decreasing diagrams technique of van Oostrom is one of the most versatile methods for establishing confluence of transition systems (abstract rewriting systems). It reduces global confluence to local confluence: a system is confluent whenever its transitions admit a locally decreasing labeling. Essentially all classical confluence criteria arise as corollaries. A central question, posed by van Oostrom in 1993, asks whether the decreasing diagrams technique is complete: Does every confluent transition system admit a locally decreasing labeling? This is Problem 56 of the RTA List of Open Problems. A positive answer was known only for countable systems, and recently up to the first uncountable cardinal $\aleph_1$. The general case remained open. We settle this thirty-three-year-old problem in full. We prove that every confluent transition system admits a locally decreasing labeling using only three labels. This bound is optimal, as two labels do not suffice even at the first uncountable cardinality. It follows that this single criterion can, in principle, certify the confluence of every confluent system, and hence of every confluent program. The entire development is machine-checked in the Isabelle/HOL and Lean proof assistants and relies only on classical logic and the axiom of choice.