Tableaux for Intuitionistic Predicate Logic as Metatheory

We show how the tableau procedure for intuitionistic predicate logic can be interpreted as the computational content of a metatheorem about Kripke models for the logic. The metatheory we use is based on constructive type theory, but we shall explore classical extensions to the theory. We begin by describing how infinite tableau developments may be represented as co-inductive types in such a theory. We then show how a bounded tableau search procedure can be interpreted as the computational content of a theorem about prefixes of (possibly infinite) Kripke countermodels. Next, we examine how classical reasoning may be introduced in the proof without affecting the computational content. This leads to a stronger theorem with the same bounded tableau search as its computational content. Finally, we discuss the unbounded tableau procedure as the computational content of a related theorem.