The self-iterability of L[E]

Let L[E] be an iterable tame extender model. We analyze to which extent L[E] knows fragments of its own iteration strategy. Specifically, we prove that inside L[E], for every cardinal K which is not a limit of Woodin cardinals there is some cutpoint t < K such that $J_k [E]$ is iterable above I with respect to iteration trees of length less than K. As an application we show L[E] to be a model of the following two cardinals versions of the diamond principle. If λ > K > a>ω1 are cardinals, then ◊$_{K.\lambda }^* $ holds true, and if in addition λ is regular, then ◊$_{K.\lambda }^* $ holds true.