  We generalize the notion of proof term to the realm of transfinite reduction.
Proof terms represent reductions in the first-order term format, thereby
facilitating their formal analysis. We show that any transfinite reduction can
be faithfully represented as an infinitary proof term, which is unique up to,
infinitary, associativity.
  Our main use of proof terms is in a definition of permutation equivalence for
transfinite reductions, on the basis of permutation equations. This definition
involves a variant of equational logic, adapted for dealing with infinite
objects.
  A proof of the compression property via proof terms is presented, which
establishes permutation equivalence between the original and the compressed
reductions.
