deletions | additions       diff --git a/The_following_theorem_is_the__.tex b/The_following_theorem_is_the__.tex  index 66722dd..714f182 100644  --- a/The_following_theorem_is_the__.tex  +++ b/The_following_theorem_is_the__.tex 
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The following theorem is the analogue in our language \begin{proof}[Proof  of the previous fact about inner product spaces.  \begin{theorem}  \label{thm:ort}  Fix $\Gamma=(\Gamma_{\mathfrak{R}}$, $\Gamma_{\mathfrak{A}})$ a pair of growth spaces of rank 2 and $\mathcal{E}$ an error space of rank 2. Assume there is $\zeta: \Nats^2 \rightarrow \Reals^{>0}$ s.t. $\zeta \in \mathcal{E}$ and $\Floor{\log \max(-\log \zeta, 2)} \in \Gamma_{\mathfrak{A}}$\footnote{If $\Floor{\log(k+2)}, \Floor{\log(j+2)} \in \Gamma_{\mathfrak{A}}$ (equivalently $\Gamma_{\text{log}}^2 \subseteq \Gamma_{\mathfrak{A}}$) then this condition holds for any $\mathcal{E}$ since we can take $\zeta = 2^{-h}$ for $h$ polynomial.}. Consider $(\mu,f)$ a distributional estimation problem and $P$ an $\mathcal{E}(\Gamma)$-optimal predictor for $(\mu,f)$. Then, $P$ is also an $\mathcal{E}^{\frac{1}{2}\sharp}(\Gamma)$-optimal predictor for $(\mu,f)$.  \end{theorem}  \begin{proof} Theorem \ref{thm:ort}]  Fix $S: \Words \xrightarrow{\Gamma} \Rats$ with $\R_S \geq \R_P$. Consider any $\sigma: \Nats^2 \rightarrow \{ \pm 1 \}$ and $n: \Nats^2 \rightarrow \Nats$. Define $t(k,j) := \sigma(k,j) 2^{-n(k,j)}$. Choose $h: \Nats^2 \rightarrow \Nats$ a polynomial s.t. $2^{-h} \in \mathcal{E}$. Then, it is easy to see there is $Q_t: \Words \xrightarrow{\Gamma} \Rats$ s.t. given $k,j \in \Nats$, $x \in \Supp \mu^k$, $y \in \WordsLen{\R_P(k,j)}$ and $z \in \WordsLen{\R_P(k,j) - \R_S(k,j)}$