I am trying to simplify the following
$ f(\alpha, \beta)= \big|1-(\frac{\alpha+\beta}{2}+\frac{|\beta-\alpha|}{2})\big|-2 \big|1-\alpha-(\frac{\alpha+\beta}{2}+\frac{|\beta-\alpha|}{2})\big|+ |\alpha-1| -\beta+|\beta-\alpha|-\frac{\alpha+\beta}{2}-\frac{|\beta-\alpha|}{2}\leq 0$
I have the following conditions: $\beta-\alpha<0$, $\alpha-1>0$
I got that the function can be simplified to
$$f(\alpha,\beta)=\big|1-\alpha \big|- 2\big|1-2\alpha\big| +\alpha -2\beta-1=0\rightarrow \alpha+\beta \geq 0$$
Is my analysis correct?
Under a different set of conditions: $\beta-\alpha<0$, $\alpha-1<0$
I have that $$f(\alpha,\beta)= \big|1-\alpha \big|- 2\big|1-2\alpha\big| + 1-2 \beta-\alpha=0 \rightarrow \beta \geq 0$$ Is my analysis for this set of conditions correct ?