I have a definition written by a professor for a simply periodic function. However, it's written so badly that I cannot understand what he means by it. To give his definition verbatim he writes the following. (He uses the reverse "in" symbol, $\ni$, to mean "such that". Hence $n_0\in N \ni \forall n\geq n_0...$ is read "$n_0$ is a natural number such that, for all $n\geq n_0$, ...)
Let $\{f_n:n\in N\}$ denote an integer-valued sequence. We say it is simply periodic if: $\exists k\in Z^+$ and $n_0\in N \ni \forall n \geq n_0$:
$$f_n\equiv f_{n+k} \mod{m}$$
and
$$f_{n+1}\equiv f_{n+k+1} \mod{m}$$
then:
$$f_0\equiv f_k \mod{m}$$
and
$$f_1\equiv f_{k+1} \mod{m}$$
Is this saying that, basically, if a sequence is simply periodic then whenever it's periodic "in the forward direction" then it's also periodic all the way back to the first terms? If my understanding is wrong, can anyone else give me a more readable definition of simply periodic? I tried searching for one and couldn't find any.