Suppose I have this recursive definition of binary strings. Let $K$ be set of binary strings. The empty string $""$ and $1$ are in $K$. If $k$ is a string in $K$, then so is $0k$, $k0$. And if $k$ is a string in $K$, then so is $k01$.
I'm trying to find a property and then prove with structural induction that is satisfied by all the strings in K and I cannot seem to find the invariant, the property that does not change as the strings in K gets more complicated. I have listed some of the strings in K below:
Length 0: $""$
Length 1: $0,1$
Length 2: $00, 01, 10$
Length 3: $000, 001, 010, 100, 101$
Length 4: $0000, 0001, 0010, 0100, 1000, 0101, 1010, 1001$
Can anyone see a pattern?
Looks to me all strings that do not have two $1$'s together.