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).

Abstract The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong–Reeb line. Two of the new models, contrary to the Harthong–Reeb line, provide accurate integer “views” on real numbers at a sequence of growing scales $B^n$ ( $B\ge2$ ).

Mechanizing proofs of geometric theorems in 3D is significantly more challenging than in 2D. As a first noteworthy case study, we consider an iconic theorem of 3D geometry: Dandelin-Gallucci's theorem. We work in the very simple but powerful framework of projective incidence geometry, where only incidence relationships are considered. We study and compare two new and very different approaches to prove this theorem. First, we propose a new proof based on the well-known Wu's method. Second, we use an original method based on matroid theory to generate a proof script which is then checked by the Coq proof assistant. For each method, we point out which parts of the proof we manage to carry out automatically and which parts are more difficult to automate and require human interaction. We hope these first developments will lead to formally proving more 3D theorems automatically and that it will be used to formally verify some key properties of computational geometry algorithms in 3D.