  We prove several results giving lower bounds for the large cardinal strength
of a failure of the singular cardinal hypothesis. The main result is the
following theorem:
  Theorem: Suppose $\kappa$ is a singular strong limit cardinal and $2^\kappa
>= \lambda$ where $\lambda$ is not the successor of a cardinal of cofinality at
most $\kappa$.
  (i) If $\cofinality(\kappa)>\gw$ then $o(\kappa)\ge\lambda$.
  (ii) If $\cofinality(\kappa)=\gw$ then either $o(\kappa)\ge\lambda$ or
$\set{\ga:K\sat o(\ga)\ge\ga^{+n}}$ is cofinal in $\kappa$ for each $n\in\gw$.
  In order to prove this theorem we give a detailed analysis of the sequences
of indiscernibles which come from applying the covering lemma to nonoverlapping
sequences of extenders.
