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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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amb_space 5
inequalities 8
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0

-1 0 0 0 1
0 -1 0 0 1
0 0 -1 0 1
0 0 0 -1 1
equations 1
-1 -1 -1 -1 2
polynomial
x[1]*x[2]*x[3]*x[4]*x[1]*x[2]*x[3]*x[4]*
(x[1]-x[2])*(x[1]-x[2])*
(x[1]-x[3])*(x[1]-x[3])*
(x[1]-x[4])*(x[1]-x[4])*
(x[2]-x[3])*(x[2]-x[3])*
(x[2]-x[4])*(x[2]-x[4])*
(x[3]-x[4])*(x[3]-x[4]);
Integral