Binary codes that do not preserve primitivity

A code $X$ is not primitivity preserving if there is a primitive list $\mathtt{ws} \in \mathtt{lists} \ X$ whose concatenation is imprimitive. We formalize a full characterization of such codes in the binary case in the proof assistant Isabelle/HOL. Part of the formalization, interesting on its own, is a description of $\{x,y\}$-interpretations of the square $xx$ if $\mathtt{length}\ y \leq \mathtt{length} \ x$. We also provide a formalized parametric solution of the related equation $x^jy^k = z^\ell$. The core of the theory is an investigation of imprimitive words which are concatenations of copies of two noncommuting words (such a pair of words is called a binary code). We follow the article [Barbin-Le Rest, Le Rest, 85] (mainly Théorème 2.1 and Lemme 3.1), while substantially optimizing the proof. See also [J.-C. Spehner. Quelques problèmes d’extension, de conjugaison et de présentation des sous-monoïdes d’un monoïde libre. PhD thesis, Université Paris VII, 1976] for an earlier result on this question, and [Maňuch, 01] for another proof.