On Equivalence Classes of Interpolation Equations

An Interpolation Equation is an equation of the form [(x)c1⋯cn=b], where c1⋯c n , b are simply typed terms containing no instantiable variable. A natural equivalence relation between two interpolation equations is the equality of their sets of solutions. We prove in this paper that given a typed variable x and a simply typed term b, the quotient by this relation of the set of all interpolation equations of the form [(x)w1⋯wp=b] contains only a finite number of classes, and relate this result to the general study of Higher Order Matching.