Sanjeev Arora and Boaz Barak show the following :
$P/poly = \cup_{c,d} DTIME (n^c)/n^d$
where $DTIME(n^c)/n^d$ is a Turing machine which is given an advice of length $O(n^d)$ and runs in $O(n^c)$ time. I do follow the proof. But I feel the proof only holds if we assume that $\forall n$ the advice given to any two $n$ length strings $x$ and $y$ is same.
But I am unable to see if the theorem still holds if the above condition if not applicable ?