What mathematical guarantees gives alpha-beta pruning? In the alpha-beta pruning version of the minimax algorithm, when one evaluates a state p with $\alpha$ and $\beta$ cutoff and gets a v value, i.e.,
v = alphabeta(p, $\alpha$, $\beta$)
are these properties true?

alphabeta(p, -$\infty$, $\beta$) = v when $\alpha$ &lt; v
alphabeta(p, $\alpha$, $\infty$) = v when v &lt; $\beta$
alphabeta(p, $\alpha$', $\beta$') = v when $\alpha$ $\le$ $\alpha$' $\le$ $\beta$' $\le$ $\beta$
if v &gt; $\beta$, then alphabeta(p, $\beta$, $\infty$) = alphabeta(p, $\alpha$, $\infty$)
if v &lt; $\alpha$, then alphabeta(p, -$\infty$, $\alpha$) = alphabeta(p, -$\infty$, $\beta$)

I've reached to this results studying the algorithm itself after reading a couple of papers. After applying it to a real case I've got an improvement of ~30% (in number of states visited, and this gives about a 30% of time execution improvement also), but I want to know if there is a mathematical background that supports these changes to the algorithm.
For a complete reference you can check An Analysis of Alpha-Beta Priming
If I am not misunderstanding eq 15 and 16 apply to your case.