Parallel postulates and decidability of intersection of lines: a mechanized study within Tarski's system of geometry

Pierre Boutry (1) , Julien Narboux (1) , Pascal Schreck (1)

1 IGG

In this paper we focus on the formalization of the proof of equivalence between different versions of Euclid's 5 th postulate. This postulate is of historical importance because for centuries many mathematicians believed that this statement was rather a theorem which could be derived from the first four of Euclid's postulates and history is rich of incorrect proofs of Euclid's 5 th postulate. These proofs are incorrect because they assume more or less implicitly a statement which is equivalent to Euclid's 5 th postulate and whose validity is taken for granted. Even though these proofs are incorrect the attempt was not pointless because the flawed proof can be turned into a proof that the unjustified statement implies the parallel postulate. In this paper we provide formal proofs verified using the Coq proof assistant that 10 different statements are equivalent to Euclid's 5 th postulate. We work in the context of Tarski's neutral geometry without continuity nor Archimedes' axiom. The formalization provide a clarification of the hypotheses used for the proofs. Following Beeson, we study the impact of the choice of a particular version of the parallel postulate on the decidability issues.

Mots clés: Euclid; parallel postulate; formalization; Tarski's geometry; Coq; foundations of geometry; decidability of intersection

Domaines: Logique en informatique [cs.LO]

Pierre Boutry, Julien Narboux, Pascal Schreck. Parallel postulates and decidability of intersection of lines: a mechanized study within Tarski's system of geometry. 2015. ⟨hal-01178236v1⟩