Consider the following functions \(\mathbf{Cost} \xrightarrow{d,u} \mathbf{Bool}\)
\(d(x>0)\mapsto false,\ d(0)\mapsto true\)
\(u(x=\infty)\mapsto false,\ d(x < \infty) \mapsto true\)
For both of these, answer:
Is it monotonic?
If so, does it satsify the monoidal monotone properties?
If so, is it strict?
The function \(d\) asks “Is it zero?”, and the function \(u\) asks “Is it finite?”.
Both of these questions are monotone: as we traverse forward in \(\leq_{Cost}\), we traverse towards being zero or being finite.
The first monoidal monotone axiom is satisifed in both because the unit (\(0\)) is mapped to the unit (\(True\)).
The second monoidal monotone axiom holds for both because addition preserves both things being zero (or not) and both things being finite (or not).
These are strict because, in \(Bool\), equality and congruence coincide.