Metamath Zero: The Cartesian Theorem Prover

TL;DR

Metamath Zero is a simple, efficient, and formally specified verification system designed for fast proof verification and serving as a trustworthy foundation for complex proof systems.

Contribution

It introduces a new verification system with a formal logic, binary proof format, and linear time verification, improving speed and reliability over existing tools.

Findings

Records fastest verification time for set.mm proofs at under 200 ms

Supports multiple proof language translations

Ensures linear time verification at scale

Abstract

As the usage of theorem prover technology expands, so too does the reliance on correctness of the tools. Metamath Zero is a verification system that aims for simplicity of logic and implementation, without compromising on efficiency of verification. It is formally specified in its own language, and supports a number of translations to and from other proof languages. This paper describes the abstract logic of Metamath Zero, essentially a multi-sorted first order logic, as well as the binary proof format and the way in which it can ensure essentially linear time verification while still being concise and efficient at scale. Metamath Zero currently holds the record for fastest verification of the $\mathtt{set .mm}$ Metamath library of proofs in ZFC (including 71 of Wiedijk's 100 formalization targets), at less than 200 ms. Ultimately, we intend to use it to verify the correctness of the implementation of the verifier down to binary executable, so it can be used as a root of trust for more complex proof systems.