Representation of Integers in Golden Mean Number System

Theorem
The positive integers $n$ are represented in the golden mean number system in their simplest form $S_n$ as follows:


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $n$ ! align="left" style = "padding: 2px 10px" | $S_n$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="left" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="left" style = "padding: 2px 10px" | $10 \cdotp 01$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="left" style = "padding: 2px 10px" | $100 \cdotp 01$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="left" style = "padding: 2px 10px" | $101 \cdotp 01$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="left" style = "padding: 2px 10px" | $1000 \cdotp 1001$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="left" style = "padding: 2px 10px" | $1010 \cdotp 0001$
 * align="right" style = "padding: 2px 10px" | $7$
 * align="left" style = "padding: 2px 10px" | $10000 \cdotp 0001$
 * align="right" style = "padding: 2px 10px" | $8$
 * align="left" style = "padding: 2px 10px" | $10001 \cdotp 0001$
 * align="right" style = "padding: 2px 10px" | $9$
 * align="left" style = "padding: 2px 10px" | $10010 \cdotp 0101$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="left" style = "padding: 2px 10px" | $10100 \cdotp 0101$
 * align="right" style = "padding: 2px 10px" | $11$
 * align="left" style = "padding: 2px 10px" | $10101 \cdotp 0101$
 * align="right" style = "padding: 2px 10px" | $12$
 * align="left" style = "padding: 2px 10px" | $100000 \cdotp 101001$
 * align="right" style = "padding: 2px 10px" | $13$
 * align="left" style = "padding: 2px 10px" | $100010 \cdotp 001001$
 * align="right" style = "padding: 2px 10px" | $14$
 * align="left" style = "padding: 2px 10px" | $100100 \cdotp 001001$
 * align="right" style = "padding: 2px 10px" | $15$
 * align="left" style = "padding: 2px 10px" | $100101 \cdotp 001001$
 * align="right" style = "padding: 2px 10px" | $16$
 * align="left" style = "padding: 2px 10px" | $101000 \cdotp 100001$
 * }
 * align="right" style = "padding: 2px 10px" | $12$
 * align="left" style = "padding: 2px 10px" | $100000 \cdotp 101001$
 * align="right" style = "padding: 2px 10px" | $13$
 * align="left" style = "padding: 2px 10px" | $100010 \cdotp 001001$
 * align="right" style = "padding: 2px 10px" | $14$
 * align="left" style = "padding: 2px 10px" | $100100 \cdotp 001001$
 * align="right" style = "padding: 2px 10px" | $15$
 * align="left" style = "padding: 2px 10px" | $100101 \cdotp 001001$
 * align="right" style = "padding: 2px 10px" | $16$
 * align="left" style = "padding: 2px 10px" | $101000 \cdotp 100001$
 * }
 * align="left" style = "padding: 2px 10px" | $100101 \cdotp 001001$
 * align="right" style = "padding: 2px 10px" | $16$
 * align="left" style = "padding: 2px 10px" | $101000 \cdotp 100001$
 * }
 * }

Proof

 * $1$ is represented by $\left[{1}\right]_\phi = \phi^0$.

From there, the algorithm for Addition of 1 in Golden Mean Number System is run.