Proof-checking Euclid

Michael Beeson (1) , Julien Narboux (2) , Freek Wiedijk (3)

1 Department of Computer Science
2 ICube - Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie
3 ICIS - Institute for Computing and Information Sciences [Nijmegen]

We used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I. We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski's formal geometry. We used proofs as close as possible to those given by Euclid, but filling Euclid's gaps and correcting errors. Euclid Book I has 48 propositions; we proved 213 theorems. The extras were partly " Book Zero " , preliminaries of a very fundamental nature, partly propositions that Euclid omitted but were used implicitly, partly advanced theorems that we found necessary to fill Euclid's gaps, and partly just variants of Euclid's propositions. We wrote these proofs in a simple fragment of first-order logic corresponding to Euclid's logic, debugged them using a custom software tool, and then checked them in the well-known and trusted proof checkers HOL Light and Coq.

Mots clés: Euclid; formalization; ITP; Coq; HOL-Light

Domaines: Logique en informatique [cs.LO]; Ingénierie assistée par ordinateur; Logiciel mathématique [cs.MS]

Michael Beeson, Julien Narboux, Freek Wiedijk. Proof-checking Euclid. 2018. ⟨hal-01612807v2⟩