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.
@InProceedings{holub_et_al:LIPIcs.ITP.2021.22, author = {Holub, \v{S}t\v{e}p\'{a}n and Starosta, \v{S}t\v{e}p\'{a}n}, title = {{Formalization of Basic Combinatorics on Words}}, booktitle = {12th International Conference on Interactive Theorem Proving (ITP 2021)}, pages = {22:1--22:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-188-7}, ISSN = {1868-8969}, year = {2021}, volume = {193}, editor = {Cohen, Liron and Kaliszyk, Cezary}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.22}, URN = {urn:nbn:de:0030-drops-139177}, doi = {10.4230/LIPIcs.ITP.2021.22}, annote = {Keywords: combinatorics on words, formalization, Isabelle/HOL} }
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