I'm reading [*A Note on a Theorem by Ladner*][1] by Balcazar and Diaz, trying to understand the proof of Theorem 2.1. I don't know if it's just old notation or if it's poorly written, but I'm having quite a bit of trouble.

 1. In step (a), is $\lambda$ the empty string?
 2. In step $(c1)$, it lists a sequence as
    $T(\lambda),T(0),T(0^2),\cdots,T(0^1)$. I don't see an obvious
    pattern defining this sequence, what is the sequence? And what's the
    difference between $T(0)$ and $T(0^1)$?
 3. In step $(c3)$, what does it mean that $T$ will "try to find" a $z$
    satisfying those properties? How does it "try" to find it?
 4. The definition of the transducer $T$ and the language $L_{m+1}$
    seems circular. Why do we not run into a chicken and egg conundrum?

  [1]: https://www.sciencedirect.com/science/article/pii/0020019082901132