Computability of a Whitney Extension

We prove the computability of a version of Whitney Extension, when the input is suitably represented. More specifically, if $F\subseteq {\mathbb{R}}^n$ is a closed set represented so that the distance function $x\mapsto d(x,F)$ can be computed, and $(f^{(\bar{k})}{)}_{\mathrm{\mid }\bar{k}\mathrm{\mid }\le m}$ is a Whitney jet of order $m$ on $F$, then we can compute $g\in C^m({\mathbb{R}}^n)$ such that $g$ and its partial derivatives coincide on $F$ with the corresponding functions of $(f^{(\bar{k})}{)}_{\mathrm{\mid }\bar{k}\mathrm{\mid }\le m}$.