Extension of Tarski's result on the decidability of reals Due to Tarski's result, it is well-known that the first-order theory of reals $(\mathbb{R},+,\cdot,&lt;,=,0,1)$ is decidable. I am working on a paper where I need an extension of this result. More precisely, I would like to consider formulas that involve rational functions, rational coefficients, minima and maxima. Trigonometric functions or exponential functions, instead, are excluded.

While this seems to me to be a straightforward consequence of Tarski's result (essentially, rewrite any formula with the above functions into one with polynomials only; e.g., rewrite $\sqrt[2]{x}$ into $\exists z (z^2 = x)$; for two polynomials $p$ and $q$, rewrite the formula $p / q \leq 1$ into $p \leq q$ and so on).

To my surprise, I could not find any paper that formally establishes this. 

Before providing the proof on my own, I wanted to ask around whether someone is aware of a paper that establishes this result.
I doubt there's a proof you can cite, since the result is obvious. The "features" you're adding are first-order definable, and first-order logic is closed under substitution.