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Due to Tarski's result, it is well-known that the first-order theory of reals $(\mathbb{R},+,\cdot,<,=,0,1)$ 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.

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  • $\begingroup$ The translation of $p/q\leq 1$ is more complex than you give, since you need to assert that $q\neq 0$ and deal with the possibility that one or both of $p$ and $q$ may be negative. $\endgroup$ – David Richerby May 21 '15 at 10:10
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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.

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