Verified Real Number Calculations: A Library for Interval Arithmetic

TL;DR

This paper introduces a formally verified library for interval arithmetic that enables automated and interactive real number calculations within theorem provers, improving the reliability of numerical proofs.

Contribution

It develops a formal, verified approach to interval arithmetic with techniques to reduce dependency effects, facilitating reliable real number calculations in proof assistants.

Findings

Formal bounds for elementary functions established

Interval splitting and Taylor series reduce dependency effects

Automated strategies for real number calculations implemented

Abstract

Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theorem-prover interaction.