Explore Lazard elimination theorems and their applications in a 42-minute lecture by Gérard Duchamp from the Institut des Hautes Etudes Scientifiques (IHES). Delve into uniform formulas for alphabets of arbitrary cardinality across groups, monoids, Lie algebras, and unital associative algebras. Examine implementable algorithms arising from this tool, focusing on k-Lie algebras, monoids, and Lie algebras. Investigate examples of iterated smash-products, emphasizing the role of string rewriting in understanding normal forms and convergence. Discover additional applications of word indexing to hyperlogarithms and character theory, based on joint work with Vu Nguyen Dinh. Gain insights from Duchamp's expertise as a researcher at LIPN UMR 7030 and IHP in this advanced mathematical exploration.