Partisan corroboration, and shifted pairing

The Herbrand theorem plays a fundamental role in automated theorem proving methods based on \emph{global variable} or \emph{rigid variable} approaches. The kernel step in procedures based on such methods can be described as the \emph{corroboration} problem (also called the \emph{Herbrand skeleton} problem), where, given a positive integer $m$, called \emph{multiplicity}, and a quantifier free formula, one seeks for a valid or provable (in classical first-order logic) disjunction of $m$ instantiations of that formula. In logic with equality this problem was recently shown to be undecidable. The first main contribution of this paper is a logical theorem, that we call the \emph{Partisan Corroboration Theorem}, that enables us to show that, for a certain interesting subclass of Horn formulas, corroboration with multiplicity one can be reduced to corroboration with any given multiplicity. The second main contribution of this paper is a \emph{finite tree automata} formalization of a technique called \emph{shifted pairing} for proving undecidability results via direct encodings of valid Turing machine computations. We call it the \emph{Shifted Pairing Theorem}. By using the Partisan Corroboration Theorem, the Shifted Pairing Theorem, and term rewriting techniques in equational reasoning, we improve upon a number of recent undecidability results related to the \emph{corroboration} problem, the \emph{simultaneous rigid E-unification} problem and the \emph{prenex fragment of intuitionistic logic with equality}