Formalization of the odd-covering and graph-certification proofs

Formalize in the Rocq proof assistant both the odd-covering criterion for abstract simplicial complexes and the correctness statements underlying vertex-edge graph certification, including the topological and homological arguments that are currently supplied only as pen-and-paper proofs.

Background

The Rocq development formally verifies the checker up to two mathematical ingredients: the odd-covering criterion used to certify completeness of a candidate vertex set, and the correctness results used to certify vertex-edge graphs. These ingredients depend on topological and homological arguments that are not available in the Rocq mathematical libraries used by the project.

The paper supplies self-contained informal proofs of both results, so the mathematical claims are established on paper but remain outside the formal proof assistant’s trusted formalization. Completing their formalization is explicitly identified as future work, and the correctness proof for graph certification is separately described as still incomplete.

References

Both involve topological and homological arguments that are not yet available in the Rocq mathematical libraries we use in our development. We therefore provide self-contained pen-and-paper proofs of these results in the present paper and leave their formalization for future work.

— Formally Certifying the Vertex Set of a Polyhedron Faster than Informal Enumeration  (2610.11913 - Allamigeon et al., 8 Oct 2026) in Section 1, Contribution; Section 5, Correctness Proof of the Certification Method; Section 6, Conclusion and Future Work