Theorem 6.1

Theorem 6.1 (Typed mirror receipt). Suppose an update selects a longest suffix occurrence $Q$ of the old input that is immediately preceded by $a$, and creates $Y=aQa$. Then $$\lambda(Y)= \begin{cases} aD_a(Q)a,&D_a(Q)\text{ defined},\\ \epsilon,&D_a(Q)\text{ undefined}. \end{cases}$$ The palindrome $\lambda(Y)$ has a complete occurrence in the old input and a pre-existing birth record. One old typed pointer can therefore install the suffix-link field of $Y$.