  We investigate non-wellfounded proof systems based on parsimonious logic, a
weaker variant of linear logic where the exponential modality ! is interpreted
as a constructor for streams over finite data. Logical consistency is
maintained at a global level by adapting a standard progressing criterion. We
present an infinitary version of cut-elimination based on finite
approximations, and we prove that, in presence of the progressing criterion, it
returns well-defined non-wellfounded proofs at its limit. Furthermore, we show
that cut-elimination preserves the progressive criterion and various regularity
conditions internalizing degrees of proof-theoretical uniformity. Finally, we
provide a denotational semantics for our systems based on the relational model.
