Formalizing Affinization of a Projective Plane in Agda

Abstract

We present a computer formalization of the problem known as affine reduction of a projective plane. The affine reduction (aka affinization) consists in the construction of an affine plane by removing a line of a projective plane. We work with a representation of von Plato axiom system of constructive geometry which allows the definition of affine and projective geometry as variants of a common structure called apartness geometry. The formalization is written in Agda, a functional programming language and proof-assistant based on the proposition-as-types paradigm. All mathematical definitions, propositions and proofs are constructed following the valid methods of constructive mathematics, and they are directly expressed in the language Agda. In addition to the description of a new formalization of an interesting mathematical problem, the paper can also contribute to introduce ideas about formalization of mathematics in type theory.