Partially Factor Lengthy Expression w/ Maple - Positive Real Numbers

[kalish1]

I need to show that the following expression,
$$a^3b-a^3c+a^3z+a^3x+a^3y-a^2bx+a^2by+a^2cx-a^2cy-a^2zx+a^2zy-a^2x^2+a^2y^2-abcz-abcx-aczx-acx^2+b^2c^2+2bc^2x+c^2x^2-b^2c-2bcx-cx^2,$$

is positive

given that:

$1.$ $\ a,b,c,x,y,z$ are positive real numbers

$2. \ \ a>b+x$

$3. \ \ c<b+y$

I know a priori that the expression is indeed positive, but I do not know how to show it, or how to use Maple to do it?

Specifically, how can I use Maple to partially factorize the expression in terms of the expressions $a-b-x$ and $c-b-y$?

Thanks for any help.

This question has been crossposted here: inequality - In Maple, how can I partially factor a lengthy symbolic expression (23 terms in 6 variables)? - Mathematics Stack Exchange

 

[Ackbach]

Interesting problem! I don't know much about Maple, but I can say that if $a-b-x$ or $c-b-y$ is a factor, then you should be able to divide your big expression by these factors, and get a simpler one. Even if it doesn't come out even, dividing the big expression by these two factors might help you out.

Mathematica might not be too helpful. If I FullSimplify the second division, I get
$$\frac{a^3 (-(b-c+x+y+z))+a^2 (x-y) (b-c+x+y+z)+a c (b+x) (x+z)-(c-1) c
(b+x)^2}{b-c+y}.$$

So, if Mathematica can't do it, I'm not certain Maple could do it, either. Mathematica is usually regarded by most as the best at symbolic manipulation.