Proof Pearl: Formalizing Spreads and Packings of the Smallest Projective Space PG(3,2) Using the Coq Proof Assistant

We formally implement the smallest three-dimensional projective space PG(3,2) in the Coq proof assistant. This projective space features 15 points and 35 lines, related by an incidence relation. We define points and lines as two plain datatypes (one with 15 constructors for points, and one with 35 constructors for lines) and the incidence relation as a boolean function, instead of using the well-known coordinate-based approach relying on GF(2)⁴. We prove that this implementation actually verifies all the usual properties of three-dimensional projective spaces. We then use an oracle to compute some characteristic subsets of objects of PG(3,2), namely spreads and packings. We formally verify that these computed objects exactly correspond to the spreads and packings of PG(3,2). For spreads, this means identifying 56 specific sets of 5 lines among 360 360 (= 15× 14× 13× 12× 11) possible ones. We then classify them, showing that the 56 spreads of PG(3,2) are all isomorphic whereas the 240 packings of PG(3,2) can be classified into two distinct classes of 120 elements. Proving these results requires partially automating the generation of some large specification files as well as some even larger proof scripts. Overall, this work can be viewed as an example of a large-scale combination of interactive and automated specifications and proofs. It is also a first step towards formalizing projective spaces of higher dimension, e.g. PG(4,2), or larger order, e.g. PG(3,3).