Implementing Geometric Algebra Products with Binary Trees

This paper presents a formalization of geometric algebras within the proof assistant Coq. We aim not only at reasoning within a theorem prover about geometric algebras but also at getting a verified implementation. This means that we take special care of providing computable definitions for all the notions that are needed in geometric algebras. In order to be able to prove formally properties of our definitions using induction, the elements of the algebra are recursively represented with binary trees. This leads to an unusual but rather concise presentation of the operations of the algebras. In this paper, we illustrate this by concentrating our presentation on the blade factorization operation in the Grassmann algebra and the different products of Clifford algebra.