If you take a subtraction-free rational identity like $(xxx+yyy)/(x+y)+xy=xx+yy$ and replace $\times$,$/$,$+$,$1$ by $+$,$-$,min,$0$, do you always get a valid min,plus,minus identity like min(min($x+x+x,y+y+y$)$-$min($x,y$),$\:x+y$)$\ =\ $min($x+x,y+y$)?



Do all subtraction-free identities tropicalize?
It suffices to show that whenever $F$ is a function $\mathbb R\_{\geq 0}^k\to\mathbb R\_{\geq 0}$ defined using $\times,/,+,1$, and $f,g$ is the corresponding tropicalization $\mathbb R^k\to\mathbb R$, for all real $x\_1,\dots,x\_k$ we have $$F(\exp(-\beta x\_1),\dots,\exp(-\beta x\_k))^{1/\beta}\to \exp(-f(x\_1,\dots,x\_k)).$$
as $\beta\to+\infty$.


This follows by structural induction on the formula defining $f$, using the following lemma.


**Lemma:** Let $F$ be one of the operations $\times,/,+$, let $f$ be the corresponding tropical operation, and let $G\_1$ and $G\_2$ be functions $\mathbb R\_{> 0}\to\mathbb R\_{> 0}$ such that
$G\_1(\beta)^{1/\beta}$ tends to some limit $e^{-x\_1}$ as $\beta\to\infty$, and likewise $G\_2(\beta)^{1/\beta}\to e^{-x\_2}$. Then
$$F(G\_1(\beta),G\_2(\beta))^{1/\beta}\to \exp(- f(x\_1,x\_2)).$$
as $\beta\to+\infty$.


**Proof:** The operations $\times$ and $/$ are easy - the only non-trivial step is
$$(G\_1(\beta)+G\_2(\beta))^{1/\beta}\to e^{-\beta \min(x\_1,x\_2)}.$$


