Rózsa: Certified Equational Reasoning

Overview & Synopsis

When does one equation imply another? Stage 2 of the SAIR Mathematics Distillation Challenge required teams to submit solvers to answer this question for sets equipped with a binary operation. For these sets, associativity, commutativity, or any other algebraic property was not assumed. Each answer required a proof or counterexample that could be verified by Lean. Programs also had to respect fixed resource limits. We present Rózsa, a Python solver that searches for both kinds of evidence. It begins with simple tests and considers more extensive searches when those fail. To find a proof, it applies a given equation and derives any further equations as needed, keeping a record of each step. To find a counterexample, it searches for operations on finite sets. It also tests one fixed infinite model. The given equation must hold for every assignment, while the target equation must fail for at least one. In either case, Rózsa turns the evidence into a Lean certificate. The submitted solver solved 199/200 problems (99.5%) in the official main evaluation getting the 6th overall position. Additional tests show that the solver can prove many implications linked by long chains of known results, but has limited success with infinite counterexamples. We explain the search methods, how they fit together, and what the unsolved cases reveal about their limits.