>**Puzzle 95.** Show that there is a monoidal monotone \\(f : \mathbf{Cost} \to \mathbf{Bool} \\) sending \\(\infty\\) to \\(\texttt{false}\\) everything else to \\(\texttt{true}\\). In economics, this means "if the cost of something is infinite, nobody can do it - otherwise it's possible, at least if you're rich enough".

Let \\(f: \mathbf{Cost} \to \mathbf{Bool}\\) be the function,
\\[
f(c) := [c > \infty],
\\]

then if \\(c = \infty\\), \\([\infty > \infty] = \texttt{false} \\) and \\(\texttt{true}\\) otherwise.

It is monoidal monotone because for all \\( c \in \mathbf{Cost} \\),
\\[
\begin{align}
\texttt{true} = I_\mathbf{Bool} \\\\
\implies f(I_\mathbf{Cost}) \\\\
= f(d(c,c)) \\\\
= f(0) \\\\
= [0 > \infty],
\end{align}
\\]

and for every \\( c_1,c_2,c_3 \in \mathbf{Cost} \\),
\\[
\begin{align}
[d(c_1,c_2) > \infty] \wedge [d(c_2,c_3) > \infty] = f(d(c_1,c_2)) \wedge f(d(c_2,c_3)) \\\\
\implies f(d(c_1,c_2)+d(c_2,c_3)) \\\\
\implies f(d(c_1,c_3)) \\\\
= [d(c_1,c_3) > \infty] .
\end{align}
\\]