I was reading subsection 5.4.6 about state machine derived variables in Mathematics for Computer Science, after defining weakly/strictly increasing and decreasing variables, this was in the footnote.
Weakly increasing variables are often also called nondecreasing. We will avoid this terminology to prevent confusion between nondecreasing variables and variables with the much weaker property of not being a decreasing variable.
so what is the difference between a "Weakly increasing variables" and "not being a decreasing variable"? and what is decreasing variable?
I don't know about the specific context, but something is usually called decreasing if it always goes downwards. That means something is "not decreasing" if it ever happens at some point that it goes upwards. However, something is nondecreasing (or weakly increasing) if it never goes downwards. That's a much stronger demand.
Said differently, something is "nondecreasing" if it never decreases, while it is "not decreasing" as long as it doesn't always decrease.