Bidirectional Interpolation for the Lambda-Calculus: Revisiting and Formalising Craig-ÄubriÄ Interpolation
arXiv:2603.03083 · doi:10.4230/LIPIcs.ITP.2026.30
The paper presents a new proof of the proof‑relevant Craig interpolation theorem for the simply‑typed lambda calculus using bidirectional typing techniques, and provides a formalisation of the result in Coq.
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 ÄubriÄ in the simply-typed lambda-calculus with sums and more recently in linear, classical and intuitionistic sequent calculi. We give a new proof of ÄubriÄ's proof-relevant interpolation theorem by building on principles of bidirectional typing, and formalise it in Rocq.