Cut Elimination for a Logic with Generic Judgments and Induction

TL;DR

This paper proves cut-elimination for an extended logic $LG^ om{ω}$ that incorporates generic judgments and induction, addressing previous limitations by relaxing restrictions on the generic quantifier $ abla$ and simplifying the logic's presentation.

Contribution

It introduces a cut-elimination proof for $LG^ om{ω}$, extending $ ext{ extbf{FOLD}}^ ext{ extbf{N}}$ with induction and simplifying its framework through equivariance.

Findings

Successfully extended the logic with induction principles.

Resolved previous modeling issues by relaxing restrictions on $ abla$.

Simplified the logic's presentation using the equivariance principle.

Abstract

This paper presents a cut-elimination proof for the logic $LG^\omega$, which is an extension of a proof system for encoding generic judgments, the logic $\FOLDNb$ of Miller and Tiu, with an induction principle. The logic $LG^\omega$, just as $\FOLDNb$, features extensions of first-order intuitionistic logic with fixed points and a ``generic quantifier'', $\nabla$, which is used to reason about the dynamics of bindings in object systems encoded in the logic. A previous attempt to extend $\FOLDNb$ with an induction principle has been unsuccessful in modeling some behaviours of bindings in inductive specifications. It turns out that this problem can be solved by relaxing some restrictions on $\nabla$, in particular by adding the axiom $B \equiv \nabla x. B$, where $x$ is not free in $B$. We show that by adopting the equivariance principle, the presentation of the extended logic can be much simplified. This paper contains the technical proofs for the results stated in \cite{tiu07entcs}; readers are encouraged to consult \cite{tiu07entcs} for motivations and examples for $LG^\omega.$