A Cyclic Proof System for HFLN

TL;DR

This paper introduces the first cyclic proof system for higher-order predicate logic with natural numbers and fixed-points, enabling inductive reasoning without explicit induction, and proves its soundness and decidability.

Contribution

It presents a novel cyclic proof system for HFLN, a higher-order logic with fixed-points, including a global condition, and establishes its soundness and decidability.

Findings

Decidability of the global cyclic proof condition

Soundness of the cyclic proof system for HFLN

A restricted completeness result for an infinitary variant

Abstract

A cyclic proof system allows us to perform inductive reasoning without explicit inductions. We propose a cyclic proof system for HFLN, which is a higher-order predicate logic with natural numbers and alternating fixed-points. Ours is the first cyclic proof system for a higher-order logic, to our knowledge. Due to the presence of higher-order predicates and alternating fixed-points, our cyclic proof system requires a more delicate global condition on cyclic proofs than the original system of Brotherston and Simpson. We prove the decidability of checking the global condition and soundness of this system, and also prove a restricted form of standard completeness for an infinitary variant of our cyclic proof system. A potential application of our cyclic proof system is semi-automated verification of higher-order programs, based on Kobayashi et al.'s recent work on reductions from program verification to HFLN validity checking.