Why acyclic holds

Imagine starting at Root and following the contents relation. Suppose you found a cyclic path and followed it. At some point, you will encounter a directory which you have already visited.

• The duplicated directory cannot be the root, since the root has no parent (and we have constrained the contents and parent relations to be reciprocal).
• The duplicated directory cannot be a non-root directory, say d, since that would mean that d has two parents (but that cannot be the case since the parent relation maps to single directory, not a set of directories).

Thus a cycle cannot exist.