Verified Quadratic Virtual Substitution for Real Arithmetic

TL;DR

This paper introduces a formally verified quadratic virtual substitution algorithm for real arithmetic in Isabelle/HOL, enabling reliable quantifier elimination for low-degree polynomial formulas and benchmarking against existing tools.

Contribution

First formalization of quadratic virtual substitution for inequalities in real arithmetic, with a modular framework and practical implementation tested on numerous benchmarks.

Findings

Verified the algorithm's correctness in Isabelle/HOL

Compared performance with existing tools on 378 benchmarks

Identified inconsistencies in some existing tools

Abstract

This paper presents a formally verified quantifier elimination (QE) algorithm for first-order real arithmetic by linear and quadratic virtual substitution (VS) in Isabelle/HOL. The Tarski-Seidenberg theorem established that the first-order logic of real arithmetic is decidable by QE. However, in practice, QE algorithms are highly complicated and often combine multiple methods for performance. VS is a practically successful method for QE that targets formulas with low-degree polynomials. To our knowledge, this is the first work to formalize VS for quadratic real arithmetic including inequalities. The proofs necessitate various contributions to the existing multivariate polynomial libraries in Isabelle/HOL. Our framework is modularized and easily expandable (to facilitate integrating future optimizations), and could serve as a basis for developing practical general-purpose QE algorithms. Further, as our formalization is designed with practicality in mind, we export our development to SML and test the resulting code on 378 benchmarks from the literature, comparing to Redlog, Z3, Wolfram Engine, and SMT-RAT. This identified inconsistencies in some tools, underscoring the significance of a verified approach for the intricacies of real arithmetic.