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?

1. *alphabeta(p, -$\infty$, $\beta$) = v* when *$\alpha$ < v*
2. *alphabeta(p, $\alpha$, $\infty$) = v* when *v < $\beta$*
3. *alphabeta(p, $\alpha$', $\beta$') = v* when *$\alpha$ $\le$ $\alpha$' $\le$ $\beta$' $\le$ $\beta$*
4. if *v > $\beta$*, then *alphabeta(p, $\beta$, $\infty$) = alphabeta(p, $\alpha$, $\infty$)*
5. if *v < $\alpha$*, 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.