# If $NTime(2^n) \subseteq DTime(n^n)$, then what can you conclude about $DSpace(n^n)$?

Assume $$NTime(2^n)\subseteq DTime(n^n)$$, what can you conclude about $$DSpace(n^n)$$?

I don't know if this is the correct approach, but here was my attempt at an answer:

Let $$A \in DSpace(n^n)$$ and define language $$B= \{ : x \in A \}$$. Define a machine $$M_B$$ that receives input $$y$$, checks it's of the form $$y= $$, and if so simulates the $$DSpace(n^n)$$ machine on $$x$$. Then $$M_B$$ decides B in $$DSpace(|x|^{|x|})=DSpace(2^{|y|})$$, and so by assumption there exists a machine $$M'_B$$ that decides $$B$$ in $$DTime(n^n)$$.

Now we can define a machine $$M'_A$$ that receives input $$x$$, builds $$y= $$, and calls $$M'_B$$. Then for input length $$n$$, $$M'_B$$ runs in time $$O(log(n^n))+O((n \cdot log(n))^{n \cdot log(n)}) = O((n \cdot log(n))^{n \cdot log(n)})$$. Since $$n \cdot log(n) = O(n^2)$$, we can say that $$M'_B$$ runs in $$O((n^2)^{n^2})=O(n^{n^2})$$. Thus, we can say that $$NTime(n^n) \subseteq DTime(n^{n^2}).$$

I still have trouble when using the padding argument in various questions (e.g., "if $$NTime/NSpace(f) \subseteq DTime/DSpace(g)...$$), so I don't know if I made a mistake or if there are stronger claims that can be made.

Also, correct me if I'm wrong, but is it true that if we have $$NTime(n) \subseteq DTime(n^c)$$ for some constant $$c$$, then $$P=NP$$? If so, then does the same conclusion hold if we have $$NTime(f(n)) \subseteq DTime(n^c)$$ for some function $$f(n)>n?$$