Bidirectional Interpolation for the Lambda-Calculus -- Revisiting and Formalising Craig-\v{C}ubri\'c Interpolation

Abstract

Craig's Interpolation theorem has a wide range of applications, from mathematical logic to computer science. Proof-theoretic techniques for establishing interpolation usually follow a method first introduced by Maehara for the Sequent Calculus and then adapted by Prawitz to Natural Deduction. The result can be strengthened to a proof-relevant version, taking proof terms into account: this was first established by \v{C}ubri\'c in the simply-typed lambda-calculus with sums and more recently in linear, classical and intuitionistic sequent calculi. We give a new proof of \v{C}ubri\'c's proof-relevant interpolation theorem by building on principles of bidirectional typing, and formalise it in Rocq.