Binary Codes that do not Preserve Primitivity

Abstract

A set of words X is not primitivity-preserving if there is a primitive list of length at least two of elements from X whose concatenation is imprimitive. Here, a word or list is primitive if it is not equal to a concatenation of several copies of a shorter word or list, and imprimitive otherwise. We formalize a full characterization of such two-element sets {x,y} in the proof assistant Isabelle/HOL. The formalization is based on an existing proof which we analyze and simplify using some innovative ideas. Part of the formalization, interesting on its own, is a description of the ways in which the square xx can appear inside a concatenation of words x and y if left| y right| le left| x right| . We also provide a formalized parametric solution of the related equation x^jy^k = z^ell .