The reverse $w^R$ of a string $w$ can be defined in two different ways:

First: Let $n=|w|.$ Then $n=|w^R|$ and AAiin$1..n,$ symbol $i$ of $w$ is the same as symbol $(n-i+1)$ of $w^R.$

Second:

$lon$ is is own reverse.

If $a$ is any symbol, then the string $a$ is its own reverse.

If $a$ is any symbol and $x$ is a string, then $\frac{\frac{?}{?}}{a\frac{x}{\frac{?}{?}}}$ is the reverse of $(\frac{\text{:}}{\frac{?}{?}}(x^{{?}^{{?}^{?}}}Ra$:$)\}.$

No string is a reverse of another unless it follows from $(1)$ through $(3)$

Prove by induction that the two definitions are equivalent.