Verified Quadratic Virtual Substitution for Real Arithmetic

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 logic of is decidable QE. However, practice, QE algorithms are highly complicated often combine multiple methods performance. VS practically successful method targets formulas with low-degree polynomials. To our knowledge, this first work to formalize including inequalities. proofs necessitate various contributions existing multivariate polynomial libraries Our framework modularized easily expandable (to facilitate integrating future optimizations), could serve as basis developing practical general-purpose algorithms. Further, formalization designed practicality mind, we export development SML test resulting code on 378 benchmarks from literature, comparing Redlog, Z3, Wolfram Engine, SMT-RAT. identified inconsistencies some tools, underscoring significance approach intricacies arithmetic.