  We develop foundations for computing Craig-Lyndon interpolants of two given
formulas with first-order theorem provers that construct clausal tableaux.
Provers that can be understood in this way include efficient machine-oriented
systems based on calculi of two families: goal-oriented such as model
elimination and the connection method, and bottom-up such as the hypertableau
calculus. We present the first interpolation method for first-order proofs
represented by closed tableaux that proceeds in two stages, similar to known
interpolation methods for resolution proofs. The first stage is an induction on
the tableau structure, which is sufficient to compute propositional
interpolants. We show that this can linearly simulate different prominent
propositional interpolation methods that operate by an induction on a
resolution deduction tree. In the second stage, interpolant lifting, quantified
variables that replace certain terms (constants and compound terms) by
variables are introduced. We justify the correctness of interpolant lifting
(for the case without built-in equality) abstractly on the basis of Herbrand's
theorem and for a different characterization of the formulas to be lifted than
in the literature. In addition, we discuss various subtle aspects that are
relevant for the investigation and practical realization of first-order
interpolation based on clausal tableaux.
