Length of Fibonacci String is Fibonacci Number

Theorem
Let $S_n$ denote the $n$th Fibonacci string.

Let $\map \len {S_n}$ denote the length of $S_n$.

Then:
 * $\map \len {S_n} = F_n$

where $F_n$ denotes the $n$th Fibonacci number.

Proof
The proof proceeds by strong induction.

For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
 * $\map \len {S_n} = F_n$

Basis for the Induction
$\map P 1$ is the case:

Thus $\map P 1$ is seen to hold.

$\map P 2$ is the case:

Thus $\map P 2$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P j$ is true, for all $j$ such that $1 \le j \le k$, then it logically follows that $\map P {k + 1}$ is true.

This is the induction hypothesis:
 * $\map \len {S_k} = F_k$

and:
 * $\map \len {S_{k - 1} } = F_{k - 1}$

from which it is to be shown that:
 * $\map \len {S_{k + 1} } = F_{k + 1}$

Induction Step
This is the induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Second Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \Z_{>0}: \map \len {S_n} = F_n$