Zeckendorf Representation of Integer shifted Left

Theorem
Let $f \left({x}\right)$ be the real function defined as:
 * $\forall x \in \R: f \left({x}\right) = \left\lfloor{x + \phi^{-1} }\right\rfloor$

where:
 * $\left\lfloor{\, \cdot \,}\right\rfloor$ denotes the floor function
 * $\phi$ denotes the golden mean.

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $n$ be expressed in Zeckendorf representation:
 * $n = F_{k_1} + F_{k_2} + \cdots + F_{k_r}$

with the appropriate restrictions on $k_1, k_2, \ldots, k_r$.

Then:
 * $F_{k_1 + 1} + F_{k_2 + 1} + \cdots + F_{k_r + 1} = f \left({\phi n}\right)$

Proof
We have:

Hence:

We have that:


 * $\hat \phi^3 + \hat \phi^5 + \hat \phi^7 + \cdots \le \hat \phi^{k_1} + \hat \phi^{k_2} + \cdots + \hat \phi^{k_r} \le \hat \phi^2 + \hat \phi^4 + \hat \phi^6 + \cdots$

Then:

Then:

Thus:
 * $\phi^{-1} - 1 \le \hat \phi^{k_1} + \hat \phi^{k_2} + \cdots + \hat \phi^{k_r} \le \phi^{-1}$

and the result follows.