Towards Automated Readable Proofs of Ruler and Compass Constructions
Vesna Marinković, Tijana Šukilović, Filip Marić
TL;DR
This work tackles the lack of readable proofs for ruler-and-compass constructions by integrating the triangle-construction solver ArgoTriCS with automated theorem provers to produce human-readable, formally checkable correctness proofs, potentially verifiable in $\text{Coq}$ or $\text Isabelle/HOL$. The authors encode construction problems and instantiated lemmas in TPTP and leverage both first-order logic provers (Vampire) and coherent-logic provers (Larus), augmenting geometric reasoning with ratio predicates to handle arithmetic. On a subset of 35 Wernick-corpus problems, Vampire proved 31 and Larus 20, illustrating the approach’s viability while highlighting the need for guiding lemmas and non-degeneracy considerations; even when proofs are readable, some cases require additional hints. The work demonstrates a feasible path toward automatically generating readable, formally verifiable proofs of geometric constructions with clear educational benefits and outlines future work to formalize lemmas in Isabelle/HOL and extend to broader geometries and degeneracy handling.
Abstract
Although there are several systems that successfully generate construction steps for ruler and compass construction problems, none of them provides readable synthetic correctness proofs for generated constructions. In the present work, we demonstrate how our triangle construction solver ArgoTriCS can cooperate with automated theorem provers for first order logic and coherent logic so that it generates construction correctness proofs, that are both human-readable and formal (can be checked by interactive theorem provers such as Coq or Isabelle/HOL). These proofs currently rely on many high-level lemmas and our goal is to have them all formally shown from the basic axioms of geometry.