Arithmetic in Metamath, Case Study: Bertrand's Postulate

TL;DR

This paper develops a system for performing arithmetic within Metamath, enabling formal proofs involving large numbers, demonstrated through the formal proof of Bertrand's postulate and discussing implications for large-scale mechanized proofs.

Contribution

It introduces a method and algorithm for arithmetic in Metamath, facilitating formal proofs with large numbers, exemplified by the proof of Bertrand's postulate.

Findings

Successfully formalized Bertrand's postulate in Metamath

Implemented arithmetic operations for large numbers in Mathematica

Discussed feasibility of large proofs like Hales' Kepler conjecture

Abstract

Unlike some other formal systems, the proof system Metamath has no built-in concept of "decimal number" in the sense that arbitrary digit strings are not recognized by the system without prior definition. We present a system of theorems and definitions and an algorithm to apply these as basic operations to perform arithmetic calculations with a number of steps proportional to an arbitrary-precision arithmetic calculation. We consider as case study the formal proof of Bertrand's postulate, which required the calculation of many small primes. Using a Mathematica implementation, we were able to complete the first formal proof in Metamath using numbers larger than 10. Applications to the mechanization of Metamath proofs are discussed, and a heuristic argument for the feasability of large proofs such as Tom Hales' proof of the Kepler conjecture is presented.