Suppose $R=Z_p[t]$ , and $\hat{R}$ its padic completion, suppose we have Endormorphism $\Phi$ of $\hat{R}$, whose redution mop p is just the absolute Frobenius of $\hat{R}/p\hat{R}$. And $R_{\infty}=dir.lim\hat{R}$, the direct limit is over copies of $\hat{R}$ indexed by the natrual numbers, and the transition maps are given by $\Phi$ mapping from one copy to the next. If we denote $\hat{R_{\infty}}$ the padic completion of $R_{\infty}$. Then is it true that $\hat{R_{\infty}}$ isomorphic to $W(\hat{R_{\infty}}/p\hat{R_{\infty}})$, the Wittvectors over $\hat{R_{\infty}}/p\hat{R_{\infty}}$ and why it is true? Thank you!
