If you take a subtractionfree 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$)?

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 nontrivial step is $$(G_1(\beta)+G_2(\beta))^{1/\beta}\to e^{\beta \min(x_1,x_2)}.$$ 


Yes. Replace $x$ with $e^{Na}$, $y$ with $e^{Nb}$, etc. Then take the log, then divide by $N$. One gets a new identity where $\times$ is replaced by $+$, $/$ by $$, $1$ by $0$, and $u+v$ by $\ln (e^{N u} + e^{N v})= \min(u,v) + \ln\left( 1+ e^{N uv}\right)/N = \min(u,v) + O(1/N)$. Then take the limit as $N$ goes to $\infty$. You now have a tropcal identity. This fits with the idea of tropical geometry as the limit of classical algebraic geometry as variables get very large. 

