I have a certain equation derived from another problem I was solving that includes floor functions. By plugging in different values I experimentally find that the following equation is true:
$\left \lfloor \frac{H-S\left ( \left \lfloor \frac{H-D\left ( K - 1\right ) - 1}{S}+1 \right \rfloor - 1\right ) - 1}{D}+1 \right \rfloor = K$
Also if I remove the floor functions, I can easily show this equality holds:
$\frac{H-S\left ( \frac{H-D\left ( K - 1\right ) - 1}{S}\right ) - 1}{D}+1 = \frac{H-\left ( H-D\left ( K - 1\right ) - 1\right ) - 1}{D}+1 = \frac{D(K-1)}{D}+1=K$
However, I don't think this is a complete proof because it is unclear to me why it is justified to just remove the floor functions. I don't really have experience solving equations with floor functions in them. Does anyone know how I would show this equality holds when not removing the floor functions?
Edit: H, S, D and K are all integers > 0. I don't have guarantees if they are odd or even.
I realize I can simplify the equation a little bit into this: $\left \lfloor \frac{H-S \left \lfloor \frac{H-D\left ( K - 1\right ) - 1}{S} \right \rfloor - 1}{D} \right \rfloor +1 = K$
Edit 2: I see now that the problem I was trying to solve is more complicated and there are many parameter combinations for which the equation doesn't hold. The equation might only be provable under certain conditions I think.