Theorem: Representation of Integers in Golden Mean Number System

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" | $101000 \cdotp 100001$
|}

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.
{{begin-eqn}}
{{eqn | n = 2
      | l = 1 + \left[{1}\right]_\phi
      | r = 1 + \left[{0 \cdotp 11}\right]_\phi
      | c = Expansion
}}
{{eqn | r = \left[{1 \cdotp 11}\right]_\phi
      | c = 
}}
{{eqn | r = \left[{10 \cdotp 01}\right]_\phi
      | c = Simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 3
      | l = 1 + \left[{10 \cdotp 01}\right]_\phi
      | r = \left[{11 \cdotp 01}\right]_\phi
      | c = 
}}
{{eqn | r = \left[{100 \cdotp 01}\right]_\phi
      | c = Simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 4
      | l = 1 + \left[{100 \cdotp 01}\right]_\phi
      | r = \left[{101 \cdotp 01}\right]_\phi
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 5
      | l = 1 + \left[{101 \cdotp 01}\right]_\phi
      | r = 1 + \left[{101 \cdotp 0011}\right]_\phi
      | c = Expansion
}}
{{eqn | r = 1 + \left[{100 \cdotp 1111}\right]_\phi
      | c = Expansion
}}
{{eqn | r = \left[{101 \cdotp 1111}\right]_\phi
      | c = 
}}
{{eqn | r = \left[{110 \cdotp 0111}\right]_\phi
      | c = Simplification
}}
{{eqn | r = \left[{1000 \cdotp 0111}\right]_\phi
      | c = Simplification
}}
{{eqn | r = \left[{1000 \cdotp 1001}\right]_\phi
      | c = Simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 6
      | l = 1 + \left[{1000 \cdotp 1001}\right]_\phi
      | r = \left[{1001 \cdotp 1001}\right]_\phi
      | c = 
}}
{{eqn | r = \left[{1010 \cdotp 0001}\right]_\phi
      | c = Simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 7
      | l = 1 + \left[{1010 \cdotp 0001}\right]_\phi
      | r = \left[{1011 \cdotp 0001}\right]_\phi
      | c = 
}}
{{eqn | r = \left[{1100 \cdotp 0001}\right]_\phi
      | c = Simplification
}}
{{eqn | r = \left[{10000 \cdotp 0001}\right]_\phi
      | c = Simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 8
      | l = 1 + \left[{10000 \cdotp 0001}\right]_\phi
      | r = \left[{10001 \cdotp 0001}\right]_\phi
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 9
      | l = 1 + \left[{10001 \cdotp 0001}\right]_\phi
      | r = 1 + \left[{10000 \cdotp 1101}\right]_\phi
      | c = Expansion
}}
{{eqn | r = \left[{10001 \cdotp 1101}\right]_\phi
      | c = 
}}
{{eqn | r = \left[{10010 \cdotp 0101}\right]_\phi
      | c = Simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 10
      | l = 1 + \left[{10010 \cdotp 0101}\right]_\phi
      | r = \left[{10011 \cdotp 0101}\right]_\phi
      | c = 
}}
{{eqn | r = \left[{10100 \cdotp 0101}\right]_\phi
      | c = Simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 11
      | l = 1 + \left[{10100 \cdotp 0101}\right]_\phi
      | r = \left[{10101 \cdotp 0101}\right]_\phi
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 12
      | l = 1 + \left[{10101 \cdotp 0101}\right]_\phi
      | r = 1 + \left[{10101 \cdotp 010011}\right]_\phi
      | c = Expansion
}}
{{eqn | r = 1 + \left[{10101 \cdotp 001111}\right]_\phi
      | c = Expansion
}}
{{eqn | r = 1 + \left[{10100 \cdotp 111111}\right]_\phi
      | c = Expansion
}}
{{eqn | r = \left[{10101 \cdotp 111111}\right]_\phi
      | c = 
}}
{{eqn | r = \left[{10110 \cdotp 011111}\right]_\phi
      | c = Simplification
}}
{{eqn | r = \left[{11000 \cdotp 011111}\right]_\phi
      | c = Simplification
}}
{{eqn | r = \left[{100000 \cdotp 011111}\right]_\phi
      | c = Simplification
}}
{{eqn | r = \left[{100000 \cdotp 100111}\right]_\phi
      | c = Simplification
}}
{{eqn | r = \left[{100000 \cdotp 101001}\right]_\phi
      | c = Simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 13
      | l = 1 + \left[{100000 \cdotp 101001}\right]_\phi
      | r = \left[{100001 \cdotp 101001}\right]_\phi
      | c = 
}}
{{eqn | r = \left[{100010 \cdotp 001001}\right]_\phi
      | c = Simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 14
      | l = 1 + \left[{100010 \cdotp 001001}\right]_\phi
      | r = \left[{100011 \cdotp 001001}\right]_\phi
      | c = 
}}
{{eqn | r = \left[{100100 \cdotp 001001}\right]_\phi
      | c = Simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 15
      | l = 1 + \left[{100100 \cdotp 001001}\right]_\phi
      | r = \left[{100101 \cdotp 001001}\right]_\phi
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 16
      | l = 1 + \left[{100101 \cdotp 001001}\right]_\phi
      | r = 1 + \left[{100100 \cdotp 111001}\right]_\phi
      | c = Expansion
}}
{{eqn | r = \left[{100101 \cdotp 111001}\right]_\phi
      | c = 
}}
{{eqn | r = \left[{100110 \cdotp 011001}\right]_\phi
      | c = Simplification
}}
{{eqn | r = \left[{101000 \cdotp 011001}\right]_\phi
      | c = Simplification
}}
{{eqn | r = \left[{101000 \cdotp 100001}\right]_\phi
      | c = Simplification
}}
{{end-eqn}}
{{qed}}