The CayleyDickson Construction in ACL2
Abstract
The CayleyDickson Construction is a generalization of the familiar construction of the complex numbers from pairs of real numbers. The complex numbers can be viewed as twodimensional vectors equipped with a multiplication. The construction can be used to construct, not only the twodimensional Complex Numbers, but also the fourdimensional Quaternions and the eightdimensional Octonions. Each of these vector spaces has a vector multiplication, v_1*v_2, that satisfies: 1. Each nonzero vector has a multiplicative inverse. 2. For the Euclidean length of a vector v, v_1 * v_2 = v_1 v2. Real numbers can also be viewed as (onedimensional) 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?
 Publication:

arXiv eprints
 Pub Date:
 May 2017
 arXiv:
 arXiv:1705.06822
 Bibcode:
 2017arXiv170506822C
 Keywords:

 Computer Science  Logic in Computer Science
 EPrint:
 In Proceedings ACL2Workshop 2017, arXiv:1705.00766