  Cyclic pre-proofs can be represented as sets of finite tree derivations with
back-links. In the frame of the first-order logic with inductive definitions,
the nodes of the tree derivations are labelled by sequents and the back-links
connect particular terminal nodes, referred to as buds, to other nodes labelled
by a same sequent. However, only some back-links can constitute sound
pre-proofs. Previously, it has been shown that special ordering and
derivability conditions, defined along the minimal cycles of the digraph
representing a particular normal form of the cyclic pre-proof, are sufficient
for validating the back-links. In that approach, a same constraint could be
checked several times when processing different minimal cycles, hence one may
require additional recording mechanisms to avoid redundant computation in order
to downgrade the time complexity to polynomial.
  We present a new approach that does not need to process minimal cycles. It
based on a normal form that allows to define the validation conditions by
taking into account only the root-bud paths from the non-singleton strongly
connected components of its digraph.
