Translation Lemma: Let f(n), g(n), h(n) be reasonable (time-constructible) functions with h(n)>n. Then
The proof uses a technique known as padding. Let L be in NTIME(f(h(n))) via a machine M. Define A by
Now if we want to compute whether x is in L we can use the DTIME(g(m)) algorithm for A on x01h(n)-|x|-1 taking total time g(h(n)) since the input has size m=h(n). QED
As an immediate consequence we get that if P=NP then E=NE (where E=DTIME(2O(n))) by letting h(n)=2n.
The translation lemma works in only one direction. You need h(n)>n, you can't unpad an arbitrary x. It's open whether E=NE implies P=NP for instance.
The lemma has many applications. Here is one example.
Theorem: If P=NP then some language in E does not have subexponential-size circuits.
Proof: In the Σ4 level of the E-time hierarchy we can compute the lexicographically-first language A that cannot be simulated by any 2n/2-size circuits. If P=NP then the polynomial-time hierarchy collapse to P. By a version of the translation lemma with h(n)=2n the polynomial-time hiearchy collapsing to P implies the E-time hierarchy collapses to E. Thus A is in E but A does not have subexponential-size circuits.