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 $$n-1$$. 

You could consider separately what happens when $n$ is an even number and when $n$ is an odd number.
If $n=2k$, then
\begin{align}
   2\left\lfloor \dfrac{n-1}{2} \right\rfloor
      &=  2\left\lfloor \dfrac{2k-1}{2} \right\rfloor\\
      &=  2\left\lfloor k - \dfrac 12 \right\rfloor\\
      &=  2(k-1) \\
      &= 2\left( \dfrac n2 - 1 \right) \\
      &= n - 2
\end{align}
If $n=2k+1$, then
\begin{align}
   2\left\lfloor \dfrac{n-1}{2} \right\rfloor
      &=  2\left\lfloor \dfrac{2k+1-1}{2} \right\rfloor\\
      &=  2\left\lfloor k \right\rfloor\\
      &=  2k \\
      &=  n - 1
\end{align}
