Automating proof search when equality is a logical connective

2026-05-19

AI Summary

Existing logical frameworks, such as λProlog and LF, struggle to support automated proof search when syntactic equality is treated as a logical connective, particularly in the presence of quantifier alternations and polarity-sensitive occurrences of equality. This work proposes a proof-search method that extends unification mechanisms to directly reason with equality as defined by its introduction rules within a first-order sequent calculus. The approach constitutes the first automated proof-search procedure for such an equality treatment, yielding a lightweight, unification-aware logical framework compatible with various first-order systems. It successfully derives core mathematical principles—including the Peano axioms without induction—thereby addressing a significant gap in equational reasoning capabilities left by current frameworks.

Abstract

Treating syntactic equality as a logical connective -- governed by left- and right-introduction rules within the sequent calculus -- offers an elegant and powerful approach to term identity. This treatment of equality allows for the derivation of core mathematical principles, such as Peano's axioms (excluding induction), and serves as a foundation for the Abella interactive proof assistant. However, integrating this equality into automated proof search remains challenging. We present a proof search procedure that extends unification to handle the complexities of quantifier alternation and equations that occur in both positive and negative occurrences. While established logical frameworks such as $λ$Prolog and LF lack direct support for this kind of equality, our procedure enables a lightweight logical framework that addresses this gap. Our system enables unification-aware proof search across a diverse range of first-order sequent calculi that can directly use this form of equality.

Problem

• proof search
• equality as logical connective
• sequent calculus
• unification
• automated reasoning

Innovation

• equality as logical connective
• unification
• sequent calculus
• automated proof search
• quantifier alternation