The Cayley-Dickson Construction in ACL2

The Cayley-Dickson Construction is a generalization of the familiar construction of the complex numbers from pairs of real numbers. The complex numbers can be viewed as two-dimensional vectors equipped with a multiplication. The construction can be used to construct, not only the two-dimensional Complex Numbers, but also the four-dimensional Quaternions and the eight-dimensional Octonions. Each of these vector spaces has a vector multiplication, v1 •v2, that satisfies: 1. Each nonzero vector, v, has a multiplicative inverse v−1. 2. For the Euclidean length of a vector |v|, |v1 •v2|= |v1| · |v2| Real numbers can also be viewed as (one-dimensional) vectors with the above two properties. ACL2(r) is used to explore this question: Given a vector space, equipped with a multiplication, satisfying the Euclidean length condition 2, given above. Make pairs of vectors into “new” vectors with a multiplication. When do the newly constructed vectors also satisfy condition 2?