Consider a proposed monoidal monotone \(\mathbf{Bool}\xrightarrow{g}\mathbf{Cost}\) with \(g(false)=\infty, g(true)=0\)
Check that the map is monotonic and that it satisfies both properties of monoidal monotones.
Is g strict?
It is monotonic: \(\forall a,b \in \mathbb{B}: a\leq b \implies g(a)\leq g(b)\)
there is only one nonidentity case in \(\mathbf{Bool}\) to cover, and it is true that \(\infty\ \leq_\mathbf{Cost}\ 0\).
Condition on units: \(0 \leq_\mathbf{Cost} g(true)\) (actually, it is equal)
In \(\mathbf{Cost}\): \(g(x) + g(y) \geq g(x \land y)\)
if both true/false, the equality condition holds.
If one is true and one is false, then LHS and RHS are \(\infty\) (as \(x \land y = False\)).
Therefore this is a strict monoidal monotone.