Updated the dyadic rational numbers and RS integrals.

src/cat/tricks/dyadic.tex

@@ -38,7 +38,13 @@
 
 \begin{proposition}
 \label{proposition:dyadic-semigroup-order}
-    Let $G$ be a commutative ordered semigroup, and $\seq{g_n} \subset G$ such that for each $n \in \natp$, $g_{n+1} + g_{n+1} \le g_n$. For each $x \in \mathbb{D} \cap [0, 1)$, let $\phi(x) = \sum_{n \in M(x)}g_n$, then
+    Let $G$ be a commutative ordered semigroup, and $\seq{g_n} \subset G$ such that 
+    \begin{enumerate}
+        \item[(a)] for each $n \in \natp$, $g_{n+1} + g_{n+1} \le g_n$.
+        \item[(b)] For each $x, y \in G$, $x + y \ge x, y$. 
+    \end{enumerate}
+    
+    For each $x \in \mathbb{D} \cap [0, 1)$, let $\phi(x) = \sum_{n \in M(x)}g_n$, then
     \begin{enumerate}
         \item For any $x, y \in \mathbb{D} \cap [0, 1)$ such that $x + y \in [0, 1)$, $\phi(x) + \phi(y) \le \phi(x + y)$.
         \item For any $x, y \in \mathbb{D} \cap [0, 1)$ with $x \le y$, $\phi(x) \le \phi(y)$.