# Inducing syntactic cut-elimination for indexed nested sequents

## TL;DR

This paper develops a syntactic method to derive cut-elimination proofs for indexed nested sequent calculi by relating them to labelled sequent formalisms, enabling new proof systems for intermediate logics.

## Contribution

It introduces a syntactic approach to induce cut-elimination for indexed nested sequents by leveraging their correspondence with labelled sequents, simplifying proof-theoretic analysis.

## Key findings

- Established correspondence between almost treelike labelled sequents and indexed nested sequents.
- Used this correspondence to derive syntactic cut-elimination proofs for indexed nested sequents.
- Presented the first indexed nested sequent calculi for intermediate logics.

## Abstract

The key to the proof-theoretic study of a logic is a proof calculus with a subformula property. Many different proof formalisms have been introduced (e.g. sequent, nested sequent, labelled sequent formalisms) in order to provide such calculi for the many logics of interest. The nested sequent formalism was recently generalised to indexed nested sequents in order to yield proof calculi with the subformula property for extensions of the modal logic K by (Lemmon-Scott) Geach axioms. The proofs of completeness and cut-elimination therein were semantic and intricate. Here we show that derivations in the labelled sequent formalism whose sequents are `almost treelike' correspond exactly to indexed nested sequents. This correspondence is exploited to induce syntactic proofs for indexed nested sequent calculi making use of the elegant proofs that exist for the labelled sequent calculi. A larger goal of this work is to demonstrate how specialising existing proof-theoretic transformations alleviate the need for independent proofs in each formalism. Such coercion can also be used to induce new cutfree calculi. We employ this to present the first indexed nested sequent calculi for intermediate logics.