A tree rewriting system for the Reflection Calculus

TL;DR

This paper introduces a tree rewriting system, ${ m TRC}$, that effectively simulates the Reflection Calculus, enabling improved proof search and formalization in proof assistants, with applications in proof theory and ordinal analysis.

Contribution

The paper presents ${ m TRC}$, a novel tree rewriting system that is complete and adequate for the Reflection Calculus, facilitating proof search and formalization.

Findings

${ m TRC}$ is complete and adequate for $ ext{f RC}$.

Normalization theorem improves proof search efficiency.

Formalization in proof assistants is facilitated.

Abstract

The $Reflection$ $Calculus$ ($\mathcal{\mathbf{RC}}$) is the fragment of the polymodal logic $\mathcal{\mathbf{GLP}}$ in the language $L^+$ whose formulas are built up from $\top$ and propositional variables using conjunction and diamond modalities. $\mathcal{\mathbf{RC}}$ is complete with respect to the arithmetical interpretation that associates modalities with reflection principles and has various applications in proof theory, specifically ordinal analysis. We present ${\sf TRC}$, a tree rewriting system that is adequate and complete with respect to $\mathcal{\mathbf{RC}}$, designed to simulate $\mathcal{\mathbf{RC}}$ derivations. ${\sf TRC}$ is based on a given correspondence between formulas of $L^{+}$ and modal trees ${\sf Tree}^{\Diamond}$. Modal trees are presented as an inductive type allowing precise positioning and transformations which give rise to the formal definition of rewriting rules and facilitates formalization in proof assistants. Furthermore, we provide a rewrite normalization theorem for systematic rule application. The normalization of the rewriting process enhances proof search efficiency and facilitates implementation. By providing ${\sf TRC}$ as an efficient provability tool for $\mathcal{\mathbf{RC}}$, we aim to help on the study of various aspects of the logic such as the subformula property and rule admissibility.