# max_symbolic and distributive law with multiplication

Minimum and maximum are distributive over multiplication of non-negative numbers, i.e., if a,b,c >= 0 then

a * max{ b, c } = max { ab, ac }

and the same works for min{}. How can I exploit this in sagemath? Consider the followiing:

```
n=5
x=var(['x_'+str(i+1) for i in range(n)])
# (x_1, x_2, x_3, x_4, x_5)
assume([x[i] >= 0 for i in range(n)])
assumptions()
# [x_1 >= 0, x_2 >= 0, x_3 >= 0, x_4 >= 0, x_5 >= 0]
```

This does **not** work as expected:

```
y = max_symbolic( x[4] * max_symbolic(x[0],x[1]), x[0] * max_symbolic(x[2],x[3]) )
# y == max(x_5*max(x_1, x_2), x_1*max(x_3, x_4))
simplify(y)
# max(x_1*max(x_3, x_4), x_5*max(x_1, x_2))
```

Simplify does not do the simplification I want it to do, which should yield

```
## max(max(x_1*x_3, x_1*x_4), max(x_5*x_1, x_5*x_2))
```

or even better

```
## max(x_1*x_3, x_1*x_4, x_5*x_1, x_5*x_2)
```

Needless to say, I have also tried `expand`

and `full_simplify`

. None of them does the trick.

So, I wonder, am I missing out on some other fancy function here, or is there a (hopefully easy) way to add a distributive law-feature to the max_symbolic function?

your simplification requires that x_1>0 and x_5>0 , so sage is right in not performing them. In principle you could use "assume(x_1>0)" but I expect that that will not make a difference. It's a tricky thing you're asking.

You are of course right, and I simply forgot to copy the corresponding lines from my worksheet. They are included now.