Reseña: A machine-checked direct proof of the Steiner-Lehmus theorem
Se ha publicado un artículo de razonamiento formalizado en Nuprl sobre geometría titulado A machine-checked direct proof of the Steiner-Lehmus theorem.
Su autora es Ariel Kellison (de Cornell University).
Su resumen es
A direct proof of the Steiner-Lehmus theorem has eluded geometers for over 170 years. The challenge has been that a proof is only considered direct if it does not rely on reductio ad absurdum. Thus, any proof that claims to be direct must show, going back to the axioms, that all of the auxiliary theorems used are also proved directly. In this paper, we give a proof of the Steiner-Lehmus theorem that is guaranteed to be direct. The evidence for this claim is derived from our methodology: we have formalized a constructive axiom set for Euclidean plane geometry in a proof assistant that implements a constructive logic and have built the proof of the Steiner-Lehmus theorem on this constructive foundation.
El trabajo se presentará en el Certified Programs and Proofs (CPP) 2022 el 18 de enero de 2022.
El código de las correspondientes teorías se encuentra aquí.
. Several generalizations of Moessner’s theorem exist. Recently, Kozen and Silva gave an algebraic proof of a general theorem that subsumes Moessner’s original theorem and its known generalizations. In this note, we describe the formalization of this theorem that the first author did in Nuprl. To the best of our knowledge, this is the first existing machine formalization. On the one hand, the formalization remains remarkably close to the original proof. On the other hand, it leads to new insights in the proof, pointing to small gaps and ambiguities that would never raise any objections in pen and pencil proofs, but which must be resolved in machine formalization.