# Consequences of the Time Hierarchy Theorem

The form of the Time Hierarchy Theorem that I have is this:

If $$f$$ is time constructible then $$\text{DTIME}(f(n)) \subsetneq \text{DTIME}(f(2n+1)^3)$$.

We want the consequences of this to be that $$\text{P} \neq \text{EXP}$$ and $$\text{EXP} \neq \text{2-EXP}$$ etc. But I can't see how this follows.

I have $$\text{P}$$ defined as $$\text{P} := \bigcup_{c=0}^\infty \text{DTIME}(n^c)$$ and $$\text{EXP}$$ as $$\text{EXP} := \bigcup_{c=0}^\infty \text{DTIME}(2^{n^c})$$.

I know that $$\text{DTIME}(n^c) \subsetneq \text{DTIME}((2n+1)^c)=\text{DTIME}(2^c(n+0.5)^c)$$. But $$c$$ is a constant, so surely the RHS is just another polynomial class.

Additionally, even if I had, say, $$\text{DTIME}(n^c) \subsetneq \text{DTIME}(2^{n^c})$$, how would I know that the limit of infinitely many of these classes still preserves the "not equal to" relation? (For instance, $$\{n\} \subsetneq \{n, n+1\}$$ but $$\bigcup_n \{n\} = \bigcup_n \{n, n+1\} = \mathbb{N}$$.)

Let $$f(n) = n^{\log n}$$. Then $$\mathsf{P} \subseteq \mathsf{DTIME}(f(n))$$ while $$\mathsf{DTIME}(f(2n+1)^3) \subseteq \mathsf{EXP}$$.
(Actually, whether this argument works depends on the exact definition of $$\mathsf{DTIME}$$. If it doesn't, take $$f_C(n) = Cn^{\log n}$$ and sum over all integer $$C>0$$.)