Iterative Fibonacci algorithm correctness proof, finding loop invariants
It kind of depends on what formalism you want to use for this proof.. But the core-idea which holds for pretty much any such formalism is the following:

Suppose you establish some invariant $I$ you will want to be able to make the following conclusion

$$
m &lt; 3 \land I \implies X
$$

where $X$ tells you that the values $i,j$ and $k$ have the right values.

You are on the right track with the argument $I = I' \cup \{ m = n \mod 3 \}$, however $I' = \{m \geq 0\}$  is way too weak to establish that $\texttt{Fib}(n)$ returns the $n$th Fibonacci number.



Try the following: 


Find some existential (for some value, eg. $z$) claim about the relationship of $n$ and $m$ (seems like you already did so)
Now use that value $z$ to show $\{ k = F_z, i = F_{z-2}, j = F_{z-1} \}$1


I hope this will help you!

1: Depends on your claim about $m$ and $n$, could also be shifted $\{ k = F_{z+2}, i = F_z, j = F_{z+1} \}$ as in gnasher729's answer which is simpler. The core idea is the same.