This question explores the context$-$free language of valid unary addition

equations. Specifically, it concerns the following language over the alphabet $\backslash$Sigma $=$

$\{1,+,=\}$:

L $=\{$a $+$ b $=$ c $|$ a$,$ b$,$ c in $1,|$a$|+|$b$|=|$c$|\}.$

Note that $|$a$|$ means the length of a$,$ and since a is all $1$s$,$ this is just the numerical

value of a$.$ For example, the string $111+11111=11111111$ is in L$,$ whereas $1+11=11$

is not. The value zero is represented by the empty string, since $|\backslash$epsi $|=0.$ That means

$+=,11+=11,$ and $+1=1$ are all valid strings in the language.