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Please, establish the above claim formally. It seems that the structure of complexity classes has so much bizarre features everywhere.

For the $\Longrightarrow$ direction, a padding technique will work out. For the other direction, the time bounds are pushed down, so hardly any current technique would succeed.

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    $\begingroup$ What did you try and where did you get stuck? We appreciate showing effort since we are not a homework service. $\endgroup$ – Juho Sep 7 '18 at 15:31
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[IKW02, Theorem 38] relativizes, due to the relativization of [IW98, Theorem 5]. The original theorem states that $\mathrm{EXP} = \mathrm{BPP} \iff \mathrm{EE} = \mathrm{BPE}$.

We have that $\mathrm{EXP}^\mathrm{ZPP} = \mathrm{BPP}^\mathrm{ZPP}\iff \mathrm{EE}^\mathrm{ZPP} = \mathrm{BPE}^\mathrm{ZPP}$.

Now by padding upward from $\mathrm{P}^\mathrm{ZPP} = \mathrm{ZPP}$, $\mathrm{BPP}^\mathrm{ZPP} = \mathrm{BPP}$, we conclude that $\mathrm{ZPEXP} = \mathrm{BPP} \iff \mathrm{ZPEE} = \mathrm{BPE}$

[IKW02] R. Impagliazzo, V. Kabanets, and A. Wigderson. In search of an easy witness: exponential time vs. probabilistic polynomial time, http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/IKW02/IKW02.pdf

[IW98] R. Impagliazzo and A. Wigderson. Randomness vs. time: De-randomization under a uniform assumption, http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/IW98/proc.pdf

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