I have a bunch of inequalities, and I'm trying to see if another inequality can be deduced from the first bunch. For example, assuming that $a \leq b$ and $c \leq d$, we can deduce that $a + c \leq b + d$, but we cannot deduce that $a + d \leq b + c$. Is there some software that allows me to take some inequalities which are assumed to be true as input, and to check if another input inequality can be deduced or proved from these earlier inequalities?
I am most familiar with MATLAB and it doesn't seem to do this. I realized that I could hack it and say for example $b = a + p$, $d = c + q$ with $p, q \geq 0$, then just writing $b + d - (a + c)$ and asking MATLAB to simplify the expression, and noticing that all the $p, q$ terms are nonnegative. But is there a more natural way to do this?