Parallel Postulates and Continuity Axioms: A Mechanized Study in Intuitionistic Logic Using Coq

Abstract

In this paper we focus on the formalization of the proofs of equivalence between different versions of Euclid's 5 th postulate. Our study is performed in the context of Tarski's neutral geometry, or equivalently in Hilbert's geometry defined by the first three groups of axioms, and uses an intuitionistic logic, assuming excluded-middle only for point equality. Our formalization provides a clarification of the conditions under which different versions of the postulates are equivalent. Following Beeson, we study which versions of the postulate are equivalent, constructively or not. We distinguish four groups of parallel postulates. In each group, the proof of their equivalence is mechanized using intuitionistic logic without continuity assumptions. For the equivalence between the groups additional assumptions are required. The equivalence between the 34 postulates is formalized in Archimedean planar neutral geometry. We also formalize a theorem due to Szmiliew. This theorem states that, assuming Archimedes' axiom, any statement which hold in the Euclidean plane and does not hold in the Hyperbolic plane is equivalent to Euclid's 5 th postulate. To obtain all these results, we have developed a large library in planar neutral geometry, including the formalization of the concept of sum of angles and the proof of the Saccheri-Legendre theorem, which states that assuming Archimedes' axiom, the sum of the angles in a triangle is at most two right angles.

Combining Pencil/Paper Proofs and Formal Proofs, A Challenge for Artificial Intelligence and Mathematics Education

“…We also assume in this chapter the postulate of alternate interior angles, stating that if two line are parallel the alternate interior angle of any secant are congruent. This postulate is equivalent to Euclid 5th postulate [Boutry et al, 2017].…”

Euclid After Computer Proof-checking

“…The axiomatization of [2], discussed in this article, is thus the last elephant in a long parade. 6 But the seal of correctness given by computer proof-checking does support the claim to be the "last elephant." That axiomatization followed the consensus of the centuries on these points: Postulate 4 should be proved, the SAS criterion of Proposition I.4 should be an axiom, and Postulate 5 has to remain.…”

“…In recent years there have been several high-profile cases of proofchecking important theorems whose large proofs involved many cases. Tarski's work on geometry, as presented in [35], has also been the subject of proof-checking and proof-finding experiments [3,4,5,6,7].…”

Proof-checking Euclid

“…However, the same cannot be said of the proofs. Many of these have problems like those of I.9 and I.7; that is, we could fix these problems only by proving some other propositions first, and the propositions of the first half of Book I had to be proved in a different order, namely 1,3,15,5,4,10,12,7,6,8,9,11, and in some cases the proofs are much more difficult than Euclid thought. After proving those early propositions, we could follow Euclid's order better, and things went well until Prop.…”