Tropical Geometry

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I would like to define a fan in SageMath using piecewise linear functions and compute its facets vertices edges and so on. For example, let P be the fan defined by

$w_{1i}=w_{23} =0, i=2,3,3,5$, $w_{24} =\min(0,x_1)$,$w_{25}= \min(0, x_1, x_1 + x_2)$, $w_{34}= x_1$, $w_{35}=\min(x_1,x_1 +x_2)$, $w_{45}= x_1+ x_2$.

It is a fan similar to the fan on page 11 of https://arxiv.org/pdf/math/0312297.pdf.

How to define the fan and compute its f-vector in SageMath? Thank you very much.