Let $(M^{4},g)$ be an ALF gravitational instanton of cyclic type. By scaling assume that the length of the circle fibres at infinity is $1$.

1. (i)

   We say that $M$ is of type $A_{k}$ for some $k\geq-1$ if there exists a compact set $K\subset M$, $R>0$ and a diffeomorphism $\phi\colon\thinspace H^{k+1}\rightarrow M\setminus K$ such that

   $$|\nabla^{l}_{g_{k+1}}(g_{k+1}-\phi^{\ast}g)|_{g_{k+1}}=O(r^{-3-l})$$

   for every $l\geq 0$.

2. (ii)

   We say that $M$ is of type $D_{m}$ for some $m\geq 0$ if there exists a compact set $K\subset M$, $R>0$ and a double cover $\phi\colon\thinspace H^{2m-4}\rightarrow M\setminus K$ such that the group $\mathbb{Z}_{2}$ of deck transformations is generated by the standard involution on $H^{2m-4}$ and

   $$|\nabla^{l}_{g_{2m-4}}(g_{2m-4}-\phi^{\ast}g)|_{g_{2m-4}}=O(r^{-3-l})$$

   for every $l\geq 0$.