  Coinduction occurs in two guises in Horn clause logic: in proofs of
self-referencing properties and relations, and in proofs involving construction
of (possibly irregular) 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's framework of uniform proofs and prove its soundness relative to
coinductive models of Horn clause logic.
