  We propose to study proof search from a coinductive point of view. In this
paper, we consider intuitionistic logic and a focused system based on
Herbelin's LJT for the implicational fragment. We introduce a variant of lambda
calculus with potentially infinitely deep terms and a means of expressing
alternatives for the description of the "solution spaces" (called B\"ohm
forests), which are a representation of all (not necessarily well-founded but
still locally well-formed) proofs of a given formula (more generally: of a
given sequent).
  As main result we obtain, for each given formula, the reduction of a
coinductive definition of the solution space to a effective coinductive
description in a finitary term calculus with a formal greatest fixed-point
operator. This reduction works in a quite direct manner for the case of Horn
formulas. For the general case, the naive extension would not even be true. We
need to study "co-contraction" of contexts (contraction bottom-up) for dealing
with the varying contexts needed beyond the Horn fragment, and we point out the
appropriate finitary calculus, where fixed-point variables are typed with
sequents. Co-contraction enters the interpretation of the formal greatest fixed
points - curiously in the semantic interpretation of fixed-point variables and
not of the fixed-point operator.
