Floor function simplification identities I can't seem to find any identity(if any)for division/multiplication involving floor functions: for example
$$\lfloor{\frac{n-1}{2}}\rfloor\cdot 2$$
I know does not simplify down to $$n-1$$. 

We have that 
$$
x = \frac{x}
{2} + \frac{x}
{2} = 2\left\lfloor {\frac{x}
{2}} \right\rfloor  + 2\left\{ {\frac{x}
{2}} \right\} = 2\left\lfloor {\frac{x}
{2}} \right\rfloor  + x\bmod 2
$$
so
$$
2\left\lfloor {\frac{x}
{2}} \right\rfloor  = x - 2\left\{ {\frac{x}
{2}} \right\} = x - x\bmod 2
$$
which means that for any $x$
$$
x - 2 < 2\left\lfloor {\frac{x}
{2}} \right\rfloor  \leqslant x
$$
