In 1998 Galatalo established an upper bound on the number of Reidemeister moves needed to convert a diagram $D$ of the unknot into a trivial loop diagram. The upper bound is a function of $n$, the number of crossings in $D$.
Galatalo reduced the problem to that of limiting the number of crossings the diagram can have during the conversion process. Specifically, he states these problems:
Problem 1. Find some function $f$ such that converting $D$ into a trivial diagram takes at most $f(n)$ Reidemeister moves.
Problem 2. Find some function $F$ such that converting $D$ into a trivial diagram never passes through an intermediate diagram with more than $F(n)$ crossings.
He claims that these problems are trivially equivalent in his footnote labelled $(\,^1)$, part of which says
If we have $F(n)$ then $f(n) < 2^{2F(n)(\log 2F(n) + 1)}$
I don't understand how a bound $F(n)$ on crossings can lead to a bound on moves, because one could do many Type III Reidemeister moves, none of which affect the number of crossings. One could bound the number of essentially different diagrams resulting from Type III moves, but that would not be as trivial as Galatalo claims. Am I missing something?
And by the way, should the bound actually be $f(n) < 2^{2F(n)(\log_2 F(n) + 1)}$? (log base 2)