  Coinduction occurs in two guises in Horn clause logic: in proofs of circular
properties and relations, and in proofs involving construction of infinite
data. Both instances of coinductive reasoning appeared in the literature
before, but a systematic analysis of these two kinds of proofs and of their
relation was lacking. We propose a general proof-theoretic framework for
handling both kinds of coinduction arising in Horn clause logic. To this aim,
we propose a coinductive extension of Miller et al framework of uniform proofs
and prove its soundness relative to coinductive models of Horn clause logic.
