  We show that the first-order theory of Sturmian words over Presburger
arithmetic is decidable. Using a general adder recognizing addition in
Ostrowski numeration systems by Baranwal, Schaeffer and Shallit, we prove that
the first-order expansions of Presburger arithmetic by a single Sturmian word
are uniformly $\omega$-automatic, and then deduce the decidability of the
theory of the class of such structures. Using an implementation of this
decision algorithm called Pecan, we automatically reprove many classical
theorems about Sturmian words in seconds, and are able to obtain new results
about antisquares and antipalindromes in characteristic Sturmian words
