Type 3 functions

Let $$A$$ be a recursive set. Let $$f$$ be a function that terminates for $$x$$ if $$x$$ belongs to $$A.$$ If $$x$$ is not in $$A,$$ then $$f(x)$$ evaluates to $$f(x + 1),$$ then $$f(x + 2),$$ and so on, until it gets to $$f(y),$$ where $$y$$ is in $$A.$$ Call functions which satisfy these properties type 3. A type 3 function always terminates, because for every $$x,$$ there must be some $$y$$ such that $$y \ge x,$$ and $$y \in A.$$ Because we're interested in termination, and not the values spit out by a function, we can omit the roots of each termination path from the call graph. This helps us focus on the more important structure in each call graph. Here's the call graph of a type 3 function, where $$A$$ is the set of all powers of 2: