Exercise 2.43#
To show this is indeed a monotone map, we must show \(x ≤_B y\) implies \(f(x) ≥_{Cost} f(y)\) for all \(x,y \in B\). There are only four cases to check:
\[\begin{split}
\begin{align} \\
false \leq false & \to \infty \geq \infty \\
false \leq true & \to \infty \geq 0 \\
true \leq false & \to 0 \geq \infty \\
true \leq true & \to 0 \geq 0 \\
\end{align}
\end{split}\]
Checking condition (a) of Definition 2.41:
\[
0 \leq g(true) = 0
\]
Checking condition (b) of Definition 2.41:
\[\begin{split}
\begin{align} \\
\infty + \infty \leq g(false) = \infty \\
\infty + 0 \leq g(false) = \infty \\
0 + \infty \leq g(false) = \infty \\
0 + 0 \leq g(true) = 0 \\
\end{align}
\end{split}\]
Yes, \(g\) is a strict monoidal monotone.