Two Non-Negative Integers have Zeckendorf Representations of which one is Shifted Representation of the Other

Theorem

Let $m, n \in \Z_{\ge 0}$ be non-negative integers.

Then there exists a unique set of integers:

$\left\{ {k_1, k_2, \ldots, k_r}\right\}$

where:

$k_1 \gg k_2 \gg \cdots \gg k_r$

where $a \gg b$ denotes that $a - b > 1$

such that:

$m = F_{k_1} + F_{k_2} + \cdots + F_{k_r}$

and:

$n = F_{k_1 + 1} + F_{k_2 + 1} + \cdots + F_{k_r + 1}$

Note that:

each of the $k$'s may be negative

and:

$r$ may equal $0$.

Proof

This theorem requires a proof.
In particular: When I have the concentration necessary. It's too hot to think at the moment.
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