Definition:Golden Mean Number System/Simplest Form

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Let $x \in \R_{\ge 0}$ have a representation $S$ in the golden mean number system.

Then $S$ is the simplest form for $x$ if and only if:

$(1): \quad$ No two adjacent $1$s appear in $S$
$(2): \quad S$ does not end with the infinite sequence $\cdotp \ldots 010101 \ldots$