By simulation I mean in the Impaglazzio-Widgerson [IW98] sense, i.e. sub-exponential deterministic simulation which appears correct i.o to every efficient adversary.
I think this is a proof: if $EXP\neq BPP$ then from [IW98] we get that BPP has such a simulation. Otherwise we have that $EXP=BPP$, which implies $RP=NP$ (because $NP \subseteq BPP$) and $EXP \in PH$. Now if $NP=RP=ZPP$ we have that $PH$ collapses to $ZPP$ and as a result $EXP$, but this cannot happen because of the assumption, so $RP\neq ZPP$ and this by the Kabanets paper "Easiness assumptions and hardness tests: trading time for zero error" implies that RP has such a simulation and as a result also NP.
This sounds like it could be a useful gap theorem. Does anyone know if it appears anywhere?
