Formal Verification of Romanov's Triplet Logic: A Verified Filter for Sliding-window 3-CNF with Application to Structured Formulas

Abstract

We present the first mechanised formalisation of Romanov's Triplet Logic (TLS) in the Rocq proof assistant. TLS is a combinatorial framework originally motivated by Boolean satisfiability, based on triplet structures and a filter that we call Simple Vertex Intersection (SVI). We formalise the core of TLS, including its translation from 3-CNF, the clearing procedure, and the SVI algorithm. For the well-formed sliding-window fragment, we prove explicit polynomial-time bounds for the filter stages and verify the translation and intersection operations. Our main contribution is a precise correctness boundary: for general formulas, SVI non-emptiness is necessary but not sufficient for satisfiability; for aligned structures, we prove a full bi-implication, extended to systems of structures. We also formalise the grouped-window translation and provide a formal counterexample to its completeness. We introduce VFR (Verified Filter for Romanov's triplet logic), an extracted OCaml prototype that implements a verified decision procedure for the sliding-window fragment and a sound filter for general 3-CNF, with a Python runtime and Docker packaging. Benchmarks corroborate the predicted behaviour, and the complete toolchain is available as a curated Zenodo artifact. The Rocq development comprises over 23,000 lines of code, with 424 proved lemmas and no unproved assumptions.