# Exercise 2.44#

To show $$d$$ is a monotone map, we must show $$x ≥_{Cost} y$$ implies $$d(x) ≤_B d(y)$$ for all $$x,y \in B$$. There are only nine cases to check, if we treat all numbers between 0 and $$\infty$$ as one of three options for each variable:

\begin{split} \begin{align} \\ 0 \geq 0 & \to T \leq T \\ 0 \geq 7 & \to T \leq F \\ 0 \geq ∞ & \to T \leq F \\ 7 \geq 0 & \to F \leq T \\ 7 \geq 7 & \to F \leq F \\ 7 \geq ∞ & \to F \leq F \\ ∞ \geq 0 & \to F \leq T \\ ∞ \geq 7 & \to F \leq F \\ ∞ \geq ∞ & \to F \leq F \\ \end{align} \end{split}

Checking condition (a) of Definition 2.41:

$T \leq d(0) = T$

Checking condition (b) of Definition 2.41:

\begin{split} \begin{align} \\ d(0) \wedge d(0) = T & \to d(0 + 0) = T \\ d(0) \wedge d(7) = F & \to d(0 + 7) = F \\ d(0) \wedge d(∞) = F & \to d(0 + ∞) = F \\ d(7) \wedge d(0) = F & \to d(7 + 0) = F \\ d(7) \wedge d(7) = F & \to d(7 + 7) = F \\ d(7) \wedge d(∞) = F & \to d(7 + ∞) = F \\ d(∞) \wedge d(0) = F & \to d(∞ + 0) = F \\ d(∞) \wedge d(7) = F & \to d(∞ + 7) = F \\ d(∞) \wedge d(∞) = F & \to d(∞ + ∞) = F \\ \end{align} \end{split}

Yes, $$d$$ is a strict monoidal monotone.

To show $$u$$ is a monotone map, we must show $$x ≥_{Cost} y$$ implies $$u(x) ≤_B u(y)$$ for all $$x,y \in B$$. There are only nine cases to check, if we treat all numbers between 0 and $$\infty$$ as one of three options for each variable:

\begin{split} \begin{align} \\ 0 \geq 0 & \to T \leq T \\ 0 \geq 7 & \to T \leq T \\ 0 \geq ∞ & \to T \leq F \\ 7 \geq 0 & \to T \leq T \\ 7 \geq 7 & \to T \leq T \\ 7 \geq ∞ & \to T \leq T \\ ∞ \geq 0 & \to F \leq T \\ ∞ \geq 7 & \to F \leq T \\ ∞ \geq ∞ & \to F \leq F \\ \end{align} \end{split}

Checking condition (a) of Definition 2.41:

$T \leq u(0) = T$

Checking condition (b) of Definition 2.41:

\begin{split} \begin{align} \\ u(0) \wedge u(0) = T & \to u(0 + 0) = T \\ u(0) \wedge u(7) = T & \to u(0 + 7) = T \\ u(0) \wedge u(∞) = F & \to u(0 + ∞) = F \\ u(7) \wedge u(0) = T & \to u(7 + 0) = T \\ u(7) \wedge u(7) = T & \to u(7 + 7) = T \\ u(7) \wedge u(∞) = F & \to u(7 + ∞) = F \\ u(∞) \wedge u(0) = F & \to u(∞ + 0) = F \\ u(∞) \wedge u(7) = F & \to u(∞ + 7) = F \\ u(∞) \wedge u(∞) = F & \to u(∞ + ∞) = F \\ \end{align} \end{split}

Yes, $$u$$ is a strict monoidal monotone.