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 $$\{x,y\}$$ { 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 $$\left| y \right| \le \left| x \right| $$ y ≤ x . We also provide a formalized parametric solution of the related equation $$x^jy^k = z^\ell $$ x j y k = z ℓ .

Combinatorics on Words is a rather young domain encompassing the study of words and formal languages. An archetypal example of a task in Combinatorics on Words is to solve the equation x ⋅ y = y ⋅ x, i.e., to describe words that commute. This contribution contains formalization of three important classical results in Isabelle/HOL. Namely i) the Periodicity Lemma (a.k.a. the theorem of Fine and Wilf), including a construction of a word proving its optimality; ii) the solution of the equation x^a ⋅ y^b = z^c with 2 ≤ a,b,c, known as the Lyndon-Schützenberger Equation; and iii) the Graph Lemma, which yields a generic upper bound on the rank of a solution of a system of equations. The formalization of those results is based on an evolving toolkit of several hundred auxiliary results which provide for smooth reasoning within more complex tasks.

We provide a reformulation and a formalization of the classical result by Juhani Karhum\"aki characterizing intersections of two languages of the form $\{x,y\}^*\cap \{u,v\}^*$. We use the terminology of morphisms which allows to formulate the result in a shorter and more transparent way, and we formalize the result in the proof assistant Isabelle/HOL.